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Name: Class: Date:

Chapter 01: Introduction

6. The fact that many children who need mental health care don’t receive it is primarily based on denial of mental health
treatment.
a. True
b. False
ANSWER: False

7. Patrick’s parents are reluctant to have him see a psychologist because they’re afraid other family members will think
they are “bad” parents. Their reluctance is most closely related to lack of resources in their local community.
a. True
b. False
ANSWER: False

8. An age-related issue of development for adolescents is the development of competence in peer relationships.
a. True
b. False
ANSWER: False

9. If a clinician wants to know if a child’s particular behavior is typical for his age and culture, she would likely be
approaching the child’s from a sociocultural perspective.
a. True
b. False
ANSWER: True

10. The belief that childhood mental health issues are overdiagnosed and overmedicated is one factor related to the stigma
of mental illness.
a. True
b. False
ANSWER: True

11. The fact that mental illness is sometimes shrouded in secrecy and rejection is one reason children and their families
may not seek treatment for mental illness.

a. True
b. False
ANSWER: True

Multiple Choice

12. The authors of this text assert that psychopathology is best understood in relationship to:
a. how children typically develop.
b. adaptation to atypical situations.
c. each child’s unique pattern of development.
d. maladaptation to environmental stressors.
ANSWER: a

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Name: Class: Date:

Chapter 01: Introduction

13. Dr. Jones tells her psychology class that it is difficult to determine one true definition of psychopathology. Which of
the following statements would best support her assertion regarding developmental psychopathology?
a. Children who experience minimal stressors may develop maladaptive behaviors.
b. Typically developing children can and do display behaviors that could be described as “abnormal.”
c. Children who display maladaptive behaviors come from a variety of situations, including socioeconomic
status.
d. Atypical development can best be defined in terms of gender expectations and age of the child.
ANSWER: b

14. A child’s behaviors might be considered typical or adaptive in one situation but not in another because these behaviors
may:
a. be rare or unusual when compared to other children of the same age or gender.
b. be considered adaptive in a given sociocultural group or situation.
c. not meet the specific criteria for a disorder as defined by the medical community.
d. be considered acceptable by psychologists who study children’s behavior.
ANSWER: b

15. At age 6, Allison’s temper tantrums are more frequent and intense than her same-aged peers. Allison’s behavior may
be considered pathological based on which of these criteria?
a. violation of sociocultural norms
b. meeting the definition of a specific mental health disorder
c. statistical deviance from peers
d. major emotional maladjustment
ANSWER: a

16. Dr. Vance works with all of his clients to identify their strengths and weaknesses and develops a plan for them to be
able to function at the highest level possible. This is known as ________ adaptation.
a. adequate
b. foundational
c. superior
d. optimal
ANSWER: d

17. Sameroff (1993) stated that “all life is characterized by disturbance that is overcome, and that only through
disturbance can we advance and grow.” How does this statement further the understanding of developmental
psychopathology?
a. It provides a basis for defining psychopathology in youth.
b. It normalizes the fact that most children can and do face challenges.
c. It creates a basis for understanding why we must provide funding for children’s mental health.
d. It supports the fact that in order to overcome adversity all humans need support.
ANSWER: b

18. The major difference between psychopathology and developmental psychopathology is the:
a. likelihood that children’s development will be negatively impacted.
b. role of culture in understanding deviant behavior.
Copyright Cengage Learning. Powered by Cognero. Page 3
Name: Class: Date:

Chapter 01: Introduction

c. understanding of how family values impact the development of maladaptive behavior.

d. belief that most deviant behavior is innate.
ANSWER: a

19. Dr. Uyenco wants to know how many new cases of autism are diagnosed each year in her particular state. Which type
of data should she access?
a. epidemiology
b. prevalence
c. identification rate
d. incidence
ANSWER: d

20. Tolan and Dodge (2005) have proposed a four-part model for a comprehensive mental health system that serves
children and their families. Based on what is known about existing barriers to services, which of the following
components would likely be a part of this model?
a. access to effective emergency care to treat crisis situations
b. preventive care provided in natural settings such as schools or daycares
c. emphasis on research-based treatments that address the cultural majority
d. an expansion of psychiatric hospitals that are equipped to handle long-term inpatient care
ANSWER: b

21. Children in Africa, Latin America, and Eastern Europe are less likely to have access to quality mental health services.
These countries may be classified as:
a. uneducated.
b. poor.
c. resource-poor.
d. rural.
ANSWER: c

22. Dr. Schwiesow, a school psychologist, is interested in understanding whether the student she is evaluating displays
off-task behaviors that are much higher than that of same-aged peers. She is likely to be considering psychopathology
from which perspective?
a. sociocultural norm
b. mental health definitions
c. statistical deviance
d. educational relevance
ANSWER: c

23. Tolan and Dodge (2005) assert that “the current state of affairs not only fails to take responsibility for the health and
welfare of children, it also fails to recognize the costs and waster in economic and human potential.” Which of the
following aspects of allocation of resources would these authors most likely promote?
a. access to care
b. financial support for research and care
c. preventative programming
d. all of the above
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Name: Class: Date:

Chapter 01: Introduction

ANSWER: d

24. According to Mukolo, Heflinger, and Wallston (2010) the stigma of mental illness includes all EXCEPT which of the
following?
a. Dimensions of stigma, including negative stereotypes
b. Actions of stigma, including hate crimes
c. Contexts of stigma, including the general public
d. Targets of stigma, including the family
ANSWER: b

Completion

25. According to the authors of this text, the understanding of developmental psychopathology must be centered in what is
_______ for a particular child’s development.
ANSWER: typical, normal

26. Dr. Wang has diagnosed Emma with separation anxiety disorder based on her presenting symptoms, but also has
considered how these symptoms impact her quality of life. Dr. Wang is using the ________ aspect of psychopathology.
ANSWER: mental health definition

27. The authors of this text believe that all children should have the support they need to reach their full potential, in spite
of challenges they may face. This is known as ________.
ANSWER: optimal adaptation

28. The example of the Hmong families and their unmarried children who continue to live at home highlights the fact that
________ are critical in understanding developmental psychopathology.
ANSWER: sociocultural norms

29. ________ are those factors that potentially prevent individuals from receiving effective treatment for mental health
issues.
ANSWER: Barriers to care

30. ________ youth are more likely to have difficulty accessing appropriate and effective treatment and prevention
programs related to mental health.
ANSWER: Minority, Disadvantaged

31. The Tolan and Dodge (2005) model of a comprehensive mental health system includes access to mental health
services, services provided in primary care settings, preventative care for high-risk children, and ________.
ANSWER: attention to cultural context and cultural competence

32. Dr. Schwiesow is studying how often schizophrenia is diagnosed in the U.S. population, as well as states where there
are more children who are diagnosed with attention deficit/hyperactivity disorder. His field of study is most likely
________.
ANSWER: ANS: developmental epidemiology

33. According to your authors the more difficult task of decreasing the stigma of mental illness is increasing society’s
_____, ______, ______ for those suffering from mental illness.

Copyright Cengage Learning. Powered by Cognero. Page 5

Name: Class: Date:

Chapter 01: Introduction

ANSWER: tolerance, compassion, understanding

34. When considering the most effective treatment approach in a country outside the United States, mental health
professionals must consider _______; ______ in order to better work within that community.
ANSWER: local and culture-based approaches; cultural norms

Subjective Short Answer

35. Discuss two factors mental health providers must consider when implementing prevention and treatments in counties
outside of the United States.
ANSWER: • Health and welfare systems
• Local and culture-based approaches
• Community caretaking and service models
• Holistic approaches

36. .Identify and discuss the dimensions of stigma identified by Mukolo, Heflinger, and Wallston.
ANSWER: o Negative stereotypes
o Devaluation
o Discrimination

Essay

SHORT ANSWER ESSAY

37. Identify the pros and cons of using the statistical deviance model of abnormality. Include a brief definition.
ANSWER: • Definition – infrequency of emotions, cognitions, and/or behaviors; too much or too little of these
• Pros – provides a point of reference for understanding psychopathology; how different is this behavior than
others their age, gender, race, etc.
• Cons – doesn’t acknowledge the role of culture and values in understanding statistical rarity; may focus of
the cultural majority

38. Define and provide an example of the sociocultural perspective on abnormality. How does this approach differ from
the statistical deviance model?
ANSWER: • Definition – behavior is viewed by how it relates to age, gender, or culture.
• Example – a child who is acting a particular way based on expectations of the family (parenting styles,
cultural expectations), neighborhood (aggression in a dangerous neighborhood), or school (value of education,
carrying weapons to protect self), etc.
• Difference – frames psychopathology in reference to a particular culture or subculture; statistical deviance
may or may not capture those differences as it generally compares to a larger, possibly more diverse group

39. Define and give an example of the mental health definition perspective on abnormality. How might this approach help
or hinder the advancement of mental health treatment for youth?
ANSWER: • Definition – psychological well-being is primary; children who are disadvantaged in terms of quality of life
or who are functioning inadequately in their current environment are at risk
• Example – an answer that identifies a particular problem a child may be experiencing (e.g., anxiety) that
would have the potential to engage in developmentally appropriate activities (e.g., going to school)
• Help or hinder – focuses on identifying those children who are at risk of developing a disorder or who are
currently experiencing problems; prevention and treatment would be emphasized
Copyright Cengage Learning. Powered by Cognero. Page 6
Name: Class: Date:

Chapter 01: Introduction

40. Compare and contrast adequate versus optimal adaptation. How might these approaches impact the outcome of youth
who are experiencing mental health issues?
ANSWER: • Adequate adaptation – functioning at a basic level that is considered sufficient for development
• Optimal adaptation – functioning at the highest level possible for that particular child
• Outcome – prevention efforts could be time-limited or sustained or include a narrow or broad range of
interventions depending on the belief about adaptation; focus of treatment could stop at a point where the child
is functional or continue until the child is more than functional; broadness of perspective which could include
only the child or the child, family, school, neighborhood

41. Compare developmental psychopathology and psychopathology in terms of basic approaches.

ANSWER: • Psychopathology – intense, frequent, and/or persistent maladaptive patterns of emotion, cognition and
behavior; could apply to adults or children; doesn’t emphasize the impact on development
• Developmental psychopathology – emphasizes how maladaptive behaviors occur in the context of typical
development and can result in short-term or long-term impairment of children; emphasizes typical and atypical
development and the concern about how it can impact current and future development

42. What is the definition of incidence and prevalence? Give examples of when it might be best to know each one of these
when working with youth with mental health issues.
ANSWER: • Incidence – current number of cases in a given population
• Prevalence – rate at which new cases are identified
• Examples – could potentially identify if a particular disorder is being diagnosed at a higher rate due to a
number of factors (e.g., environmental toxins, revised definitions, etc.); can compare prevalence to incidence
to identify trends; can identify pockets of different disorders which could lead to possible etiology and where
to focus prevention and treatment efforts

43. What are two problems related to estimating rates of disorders in childhood? How might researchers guard against
these potential problems?
ANSWER: • Two problems – may underestimate actual rates of disorders because numbers are usually based on random
sampling of a population; may not capture relevant factors such as ethnicity, gender, SES
• Researchers – increase number in sample; compare numbers to other studies; repeat studies over time;
identify relevant factors (e.g., gender, ethnicity) and include those in the sample

44. Identify the barriers to care that minority and socioeconomically disadvantaged youth might face. How and why do
these barriers differ from other youth?
ANSWER: • Barriers – structural (transportation, inability to pay, lack of competent providers); perception (cultural
norms that place a stigma on seeking mental health treatment based on lack of education); preventative (lack
of programs available in neighborhood, community or schools because of funding issues)
• Different from other youth – related to education, insurance benefits, higher quality schools, understanding
of cultural norms related to socio-economic status

45. Using what is known about developmental psychopathology, develop a brief public service announcement (3-4
sentences) highlighting the important aspects and/or misconceptions related to mental health issues in youth and what can
and should be done to help these youth.
ANSWER: • Any reasonable combination of the following:
o Definition of developmental psychopathology
o Prevalence of mental health issues in youth
o Stigma related to psychopathology
o Potential for harm to child and society – loss of productivity, cost, etc.
o What we know about what can be done to help children – preventative care and treatment
Copyright Cengage Learning. Powered by Cognero. Page 7
Name: Class: Date:

Chapter 01: Introduction

LONG ANSWER ESSAY

46. According to the authors of this text, why is it important to maintain a flexible and changing understanding of a child’s
strengths and weaknesses?
ANSWER: • Includes a wide variety of factors related to a child’s development
• Identifies how many factors could impact a child’s development including culture, society, family, ethnicity
• Children change and develop over time and have different needs over time
• Focuses on both challenges and strengths that can be a part of prevention and treatment

47. A 15-year-old freshman has been referred to the school psychologist by her mother. The concerns include poor grades,
withdrawal from her family, increased moodiness, and conflicts with peers. Briefly describe how this school psychologist
might view this child’s problems from a statistical deviance perspective, sociocultural perspective, and mental health
perspective.
ANSWER: • Statistical deviance – how different is her behavior from others her age and gender
• Sociocultural perspective – what the cultural expectations of her are in terms of her family, neighborhood,
school, ethnicity
• Mental health perspective – how her behavior is impacting her functioning in various environments (e.g.,
home, school) and what the potential is for future harm

48. What are three key issues related to the stigma of psychopathology in children, and why are they important to address?
For each of these issues, propose one way to combat stigma related to psychopathology in children.
ANSWER: • Issues – any issues related to personal, familial, social, and institutional stigma
• Combat through education, prevention programs in primary settings (schools, doctor’s offices, etc.), public
service announcements, and access to effective treatment

49. Based on what you’ve read in the text, develop a lobbying strategy to present to policy makers regarding key
components of a supportive and effective mental health care policy for youth.
ANSWER: • Key components – access to preventative programs for all children and families; may emphasize the cost of
lost productivity if society doesn’t address children’s issues; access to quality and affordable treatment;
combating misconceptions about mental health in youth; financial support for quality programs; education of
those who come in contact with children – pediatricians, educators, etc. Could also reference the Tolan and
Dodge (2005) model – access to mental quality mental health services, prevention provided in primary care
settings, focus on high-risk families and children, and emphasis on cultural context and competence.

50. Define and give an example of stigma related to psychopathology in children. In what ways does today’s media help
or hurt the stigma related to mental health?
ANSWER: • Any issues related to personal, familial, social and institutional stigma
• Reference to music, TV, movies, etc., that either educate, stereotype, or stigmatize mental health issues in
children

51. Based on what you’ve learned about developmental psychopathology, develop a preventative program that would be
housed in the local school district. Identify at least four key components to such a program.
ANSWER: • Any reasonable response that includes prevention, education, efforts to reduce stigma, access to quality
professionals and interventions, early recognition of mental health issues, referral to effective and culturally
competent professionals; also should include a program that address the whole child (cognitive, social, etc.)
• Could also reference the Tolan and Dodge (2005) model – access to mental quality mental health services,
prevention provided in primary care settings, focus on high-risk families and children, and emphasis on
cultural context and competence

Copyright Cengage Learning. Powered by Cognero. Page 8

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use these to test the rule would be a vicious circle.
 Chapter X
THE NEW QUANTUM THEORY
The conflict between quantum theory and classical theory becomes
especially acute in the problem of the propagation of light. Here in
effect it becomes a conflict between the corpuscular theory of light
and the wave theory.
In the early days it was often asked, How large is a quantum of
light? One answer is obtained by examining a star image formed
with the great 100-inch reflector at Mt. Wilson. The diffraction pattern
shows that each emission from each atom must be filling the whole
mirror. For if one atom illuminates one part only and another atom
another part only, we ought to get the same effect by illuminating
different parts of the mirror by different stars (since there is no
particular virtue in using atoms from the same star); actually the
diffraction pattern then obtained is not the same. The quantum must
be large enough to cover a 100-inch mirror.
But if this same star-light without any artificial concentration falls
on a film of potassium, electrons will fly out each with the whole
energy of a quantum. This is not a trigger action releasing energy
already stored in the atom, because the amount of energy is fixed by
the nature of the light, not by the nature of the atom. A whole
quantum of light energy must have gone into the atom and blasted
away the electron. The quantum must be small enough to enter an
atom.
I do not think there is much doubt as to the ultimate origin of this
contradiction. We must not think about space and time in connection
with an individual quantum; and the extension of a quantum in space
has no real meaning. To apply these conceptions to a single
quantum is like reading the Riot Act to one man. A single quantum
has not travelled 50 billion miles from Sirius; it has not been 8 years
on the way. But when enough quanta are gathered to form a quorum
there will be found among them statistical properties which are the
genesis of the 50 billion miles’ distance of Sirius and the 8 years’
journey of the light.

Wave-Theory of Matter. It is comparatively easy to realise what we

have got to do. It is much more difficult to start to do it. Before we
review the attempts in the last year or two to grapple with this
problem we shall briefly consider a less drastic method of progress
initiated by De Broglie. For the moment we shall be content to accept
the mystery as a mystery. Light, we will say, is an entity with the
wave property of spreading out to fill the largest object glass and
with all the well-known properties of diffraction and interference;
simultaneously it is an entity with the corpuscular or bullet property of
expending its whole energy on one very small target. We can
scarcely describe such an entity as a wave or as a particle; perhaps
as a compromise we had better call it a “wavicle”.
There is nothing new under the sun, and this latest volte-face
almost brings us back to Newton’s theory of light—a curious mixture
of corpuscular and wave-theory. There is perhaps a pleasing
sentiment in this “return to Newton”. But to suppose that Newton’s
scientific reputation is especially vindicated by De Broglie’s theory of
light, is as absurd as to suppose that it is shattered by Einstein’s
theory of gravitation. There was no phenomenon known to Newton
which could not be amply covered by the wave-theory; and the
clearing away of false evidence for a partly corpuscular theory, which
influenced Newton, is as much a part of scientific progress as the
bringing forward of the (possibly) true evidence, which influences us
to-day. To imagine that Newton’s great scientific reputation is tossing
up and down in these latter-day revolutions is to confuse science
with omniscience.
To return to the wavicle.—If that which we have commonly
regarded as a wave partakes also of the nature of a particle, may not
that which we have commonly regarded as a particle partake also of
the nature of a wave? It was not until the present century that
experiments were tried of a kind suitable to bring out the corpuscular
aspect of the nature of light; perhaps experiments may still be
possible which will bring out a wave aspect of the nature of an
electron.
So, as a first step, instead of trying to clear up the mystery we try
to extend it. Instead of explaining how anything can possess
simultaneously the incongruous properties of wave and particle we
seek to show experimentally that these properties are universally
associated. There are no pure waves and no pure particles.
The characteristic of a wave-theory is the spreading of a ray of
light after passing through a narrow aperture—a well-known
phenomenon called diffraction. The scale of the phenomenon is
proportional to the wave-length of the light. De Broglie has shown us
how to calculate the lengths of the waves (if any) associated with an
electron, i.e. considering it to be no longer a pure particle but a
wavicle. It appears that in some circumstances the scale of the
corresponding diffraction effects will not be too small for
experimental detection. There are now a number of experimental
results quoted as verifying this prediction. I scarcely know whether
they are yet to be considered conclusive, but there does seem to be
serious evidence that in the scattering of electrons by atoms
phenomena occur which would not be produced according to the
usual theory that electrons are purely corpuscular. These effects
analogous to the diffraction and interference of light carry us into the
stronghold of the wave-theory. Long ago such phenomena ruled out
all purely corpuscular theories of light; perhaps to-day we are finding
similar phenomena which will rule out all purely corpuscular theories
of matter.[33]
A similar idea was entertained in a “new statistical mechanics”
developed by Einstein and Bose—at least that seems to be the
physical interpretation of the highly abstract mathematics of their
theory. As so often happens the change from the classical
mechanics, though far-reaching in principle, gave only insignificant
corrections when applied to ordinary practical problems. Significant
differences could only be expected in matter much denser than
anything yet discovered or imagined. Strange to say, just about the
time when it was realised that very dense matter might have strange
properties different from those expected according to classical
conceptions, very dense matter was found in the universe.
Astronomical evidence seems to leave practically no doubt that in
the so-called white dwarf stars the density of matter far transcends
anything of which we have terrestrial experience; in the Companion
of Sirius, for example, the density is about a ton to the cubic inch.
This condition is explained by the fact that the high temperature and
correspondingly intense agitation of the material breaks up (ionises)
the outer electron systems of the atoms, so that the fragments can
be packed much more closely together. At ordinary temperatures the
minute nucleus of the atom is guarded by outposts of sentinel
electrons which ward off other atoms from close approach even
under the highest pressures; but at stellar temperatures the agitation
is so great that the electrons leave their posts and run all over the
place. Exceedingly tight packing then becomes possible under high
enough pressure. R. H. Fowler has found that in the white dwarf
stars the density is so great that classical methods are inadequate
and the new statistical mechanics must be used. In particular he has
in this way relieved an anxiety which had been felt as to their
ultimate fate; under classical laws they seemed to be heading
towards an intolerable situation—the star could not stop losing heat,
but it would have insufficient energy to be able to cool down![34]

Transition to a New Theory. By 1925 the machinery of current theory

had developed another flaw and was urgently calling for
reconstruction; Bohr’s model of the atom had quite definitely broken
down. This is the model, now very familiar, which pictures the atom
as a kind of solar system with a central positively charged nucleus
and a number of elecrons describing orbits about it like planets, the
important feature being that the possible orbits are limited by the
rules referred to on p. 190. Since each line in the spectrum of the
atom is emitted by the jump of an electron between two particular
orbits, the classification of the spectral lines must run parallel with
the classification of the orbits by their quantum numbers in the
model. When the spectroscopists started to unravel the various
series of lines in the spectra they found it possible to assign an orbit
jump for every line—they could say what each line meant in terms of
the model. But now questions of finer detail have arisen for which
this correspondence ceases to hold. One must not expect too much
from a model, and it would have been no surprise if the model had
failed to exhibit minor phenomena or if its accuracy had proved
imperfect. But the kind of trouble now arising was that only two orbit
jumps were provided in the model to represent three obviously
associated spectral lines; and so on. The model which had been so
helpful in the interpretation of spectra up to a point, suddenly
became altogether misleading; and spectroscopists were forced to
turn away from the model and complete their classification of lines in
a way which ignored it. They continued to speak of orbits and orbit
jumps but there was no longer a complete one-to-one
correspondence with the orbits shown in the model.[35]
The time was evidently ripe for the birth of a new theory. The
situation then prevailing may be summarised as follows:
(1) The general working rule was to employ the classical laws
with the supplementary proviso that whenever anything of the nature
of action appears it must be made equal to , or sometimes to an
integral multiple of .
(2) The proviso often led to a self-contradictory use of the
classical theory. Thus in the Bohr atom the acceleration of the
electron in its orbit would be governed by classical electrodynamics
whilst its radiation would be governed by the rule. But in classical
electrodynamics the acceleration and the radiation are indissolubly
connected.
(3) The proper sphere of classical laws was known. They are a
form taken by the more general laws in a limiting case, viz. when the
number of quanta concerned is very large. Progress in the
investigation of the complete system of more general laws must not
be hampered by classical conceptions which contemplate only the
limiting case.
(4) The present compromise involved the recognition that light
has both corpuscular and wave properties. The same idea seems to
have been successfully extended to matter and confirmed by
experiment. But this success only renders the more urgent some
less contradictory way of conceiving these properties.
(5) Although the above working rule had generally been
successful in its predictions, it was found to give a distribution of
electron orbits in the atom differing in some essential respects from
that deduced spectroscopically. Thus a reconstruction was required
not only to remove logical objections but to meet the urgent
demands of practical physics.

Development of the New Quantum Theory. The “New Quantum

Theory” originated in a remarkable paper by Heisenberg in the
autumn of 1925. I am writing the first draft of this lecture just twelve
months after the appearance of the paper. That does not give long
for development; nevertheless the theory has already gone through
three distinct phases associated with the names of Born and Jordan,
Dirac, Schrödinger. My chief anxiety at the moment is lest another
phase of reinterpretation should be reached before the lecture can
be delivered. In an ordinary way we should describe the three
phases as three distinct theories. The pioneer work of Heisenberg
governs the whole, but the three theories show wide differences of
thought. The first entered on the new road in a rather matter-of-fact
way; the second was highly transcendental, almost mystical; the
third seemed at first to contain a reaction towards classical ideas, but
that was probably a false impression. You will realise the anarchy of
this branch of physics when three successive pretenders seize the
throne in twelve months; but you will not realise the steady progress
made in that time unless you turn to the mathematics of the subject.
As regards philosophical ideas the three theories are poles apart; as
regards mathematical content they are one and the same.
Unfortunately the mathematical content is just what I am forbidden to
treat of in these lectures.
I am, however, going to transgress to the extent of writing down
one mathematical formula for you to contemplate; I shall not be so
unreasonable as to expect you to understand it. All authorities seem
to be agreed that at, or nearly at, the root of everything in the
physical world lies the mystic formula

We do not yet understand that; probably if we could understand it we

should not think it so fundamental. Where the trained mathematician
has the advantage is that he can use it, and in the past year or two it
has been used in physics with very great advantage indeed. It leads
not only to those phenomena described by the older quantum laws
such as the rule, but to many related phenomena which the older
formulation could not treat.
On the right-hand side, besides (the atom of action) and the
merely numerical factor , there appears (the square root of -1)
which may seem rather mystical. But this is only a well-known
subterfuge; and far back in the last century physicists and engineers
were well aware that in their formulae was a kind of signal to
look out for waves or oscillations. The right-hand side contains
nothing unusual, but the left-hand side baffles imagination. We call
and co-ordinates and momenta, borrowing our vocabulary from the
world of space and time and other coarse-grained experience; but
that gives no real light on their nature, nor does it explain why is
so ill-behaved as to be unequal to .
It is here that the three theories differ most essentially. Obviously
and cannot represent simple numerical measures, for then
would be zero. For Schrödinger is an operator. His
“momentum” is not a quantity but a signal to us to perform a certain
mathematical operation on any quantities which may follow. For Born
and Jordan is a matrix—not one quantity, nor several quantities,
but an infinite number of quantities arranged in systematic array. For
Dirac is a symbol without any kind of numerical interpretation; he
calls it a -number, which is a way of saying that it is not a number at
all.
I venture to think that there is an idea implied in Dirac’s treatment
which may have great philosophical significance, independently of
any question of success in this particular application. The idea is that
in digging deeper and deeper into that which lies at the base of
physical phenomena we must be prepared to come to entities which,
like many things in our conscious experience, are not measurable by
numbers in any way; and further it suggests how exact science, that
is to say the science of phenomena correlated to measure-numbers,
can be founded on such a basis.
One of the greatest changes in physics between the nineteenth
century and the present day has been the change in our ideal of
scientific explanation. It was the boast of the Victorian physicist that
he would not claim to understand a thing until he could make a
model of it; and by a model he meant something constructed of
levers, geared wheels, squirts, or other appliances familiar to an
engineer. Nature in building the universe was supposed to be
dependent on just the same kind of resources as any human
mechanic; and when the physicist sought an explanation of
phenomena his ear was straining to catch the hum of machinery.
The man who could make gravitation out of cog-wheels would have
been a hero in the Victorian age.
Nowadays we do not encourage the engineer to build the world
for us out of his material, but we turn to the mathematician to build it
out of his material. Doubtless the mathematician is a loftier being
than the engineer, but perhaps even he ought not to be entrusted
with the Creation unreservedly. We are dealing in physics with a
symbolic world, and we can scarcely avoid employing the
mathematician who is the professional wielder of symbols; but he
must rise to the full opportunities of the responsible task entrusted to
him and not indulge too freely his own bias for symbols with an
arithmetical interpretation. If we are to discern controlling laws of
Nature not dictated by the mind it would seem necessary to escape
as far as possible from the cut-and-dried framework into which the
mind is so ready to force everything that it experiences.
I think that in principle Dirac’s method asserts this kind of
emancipation. He starts with basal entities inexpressible by numbers
or number-systems and his basal laws are symbolic expressions
unconnected with arithmetical operations. The fascinating point is
that as the development proceeds actual numbers are exuded from
the symbols. Thus although and individually have no arithmetical
interpretation, the combination has the arithmetical
interpretation expressed by the formula above quoted. By furnishing
numbers, though itself non-numerical, such a theory can well be the
basis for the measure-numbers studied in exact science. The
measure-numbers, which are all that we glean from a physical
survey of the world, cannot be the whole world; they may not even
be so much of it as to constitute a self-governing unit. This seems
the natural interpretation of Dirac’s procedure in seeking the
governing laws of exact science in a non-arithmetical calculus.
I am afraid it is a long shot to predict anything like this emerging
from Dirac’s beginning; and for the moment Schrödinger has rent
much of the mystery from the ’s and ’s by showing that a less
transcendental interpretation is adequate for present applications.
But I like to think that we may have not yet heard the last of the idea.
Schrödinger’s theory is now enjoying the full tide of popularity,
partly because of intrinsic merit, but also, I suspect, partly because it
is the only one of the three that is simple enough to be
misunderstood. Rather against my better judgment I will try to give a
rough impression of the theory. It would probably be wiser to nail up
over the door of the new quantum theory a notice, “Structural
alterations in progress—No admittance except on business”, and
particularly to warn the doorkeeper to keep out prying philosophers. I
will, however, content myself with the protest that, whilst
Schrödinger’s theory is guiding us to sound and rapid progress in
many of the mathematical problems confronting us and is
indispensable in its practical utility, I do not see the least likelihood
that his ideas will survive long in their present form.

Outline of Schrödinger’s Theory. Imagine a sub-aether whose

surface is covered with ripples. The oscillations of the ripples are a
million times faster than those of visible light—too fast to come within
the scope of our gross experience. Individual ripples are beyond our
ken; what we can appreciate is a combined effect—when by
convergence and coalescence the waves conspire to create a
disturbed area of extent large compared with individual ripples but
small from our own Brobdingnagian point of view. Such a disturbed
area is recognised as a material particle; in particular it can be an
electron.
The sub-aether is a dispersive medium, that is to say the ripples
do not all travel with the same velocity; like water-ripples their speed
depends on their wave-length or period. Those of shorter period
travel faster. Moreover the speed may be modified by local
conditions. This modification is the counterpart in Schrödinger’s
theory of a field of force in classical physics. It will readily be
understood that if we are to reduce all phenomena to a propagation
of waves, then the influence of a body on phenomena in its
neighbourhood (commonly described as the field of force caused by
its presence) must consist in a modification of the propagation of
waves in the region surrounding it.
We have to connect these phenomena in the sub-aether with
phenomena in the plane of our gross experience. As already stated,
a local stormy region is detected by us as a particle; to this we now
add that the frequency (number of oscillations per second) of the
waves constituting the disturbance is recognised by us as the energy
of the particle. We shall presently try to explain how the period
manages to manifest itself to us in this curiously camouflaged way;
but however it comes about, the recognition of a frequency in the
sub-aether as an energy in gross experience gives at once the
constant relation between period and energy which we have called
the rule.
Generally the oscillations in the sub-aether are too rapid for us to
detect directly; their frequency reaches the plane of ordinary
experience by affecting the speed of propagation, because the
speed depends (as already stated) on the wave-length or frequency.
Calling the frequency , the equation expressing the law of
propagation of the ripples will contain a term in . There will be
another term expressing the modification caused by the “field of
force” emanating from the bodies present in the neighbourhood. This
can be treated as a kind of spurious , since it emerges into our
gross experience by the same method that does. If produces
those phenomena which make us recognise it as energy, the
spurious will produce similar phenomena corresponding to a
spurious kind of energy. Clearly the latter will be what we call
potential energy, since it originates from influences attributable to the
presence of surrounding objects.
Assuming that we know both the real and the spurious or
potential for our ripples, the equation of wave-propagation is
settled, and we can proceed to solve any problem concerning wave-
propagation. In particular we can solve the problem as to how the
stormy areas move about. This gives a remarkable result which
provides the first check on our theory. The stormy areas (if small
enough) move under precisely the same laws that govern the
motions of particles in classical mechanics. The equations for the
motion of a wave-group with given frequency and potential frequency
are the same as the classical equations of motion of a particle with
the corresponding energy and potential energy.
It has to be noticed that the velocity of a stormy area or group of
waves is not the same as the velocity of an individual wave. This is
well known in the study of water-waves as the distinction between
group-velocity and wave-velocity. It is the group-velocity that is
observed by us as the motion of the material particle.
We should have gained very little if our theory did no more than
re-establish the results of classical mechanics on this rather fantastic
basis. Its distinctive merits begin to be apparent when we deal with
phenomena not covered by classical mechanics. We have
considered a stormy area of so small extent that its position is as
definite as that of a classical particle, but we may also consider an
area of wider extent. No precise delimitation can be drawn between
a large area and a small area, so that we shall continue to associate
the idea of a particle with it; but whereas a small concentrated storm
fixes the position of the particle closely, a more extended storm
leaves it very vague. If we try to interpret an extended wave-group in
classical language we say that it is a particle which is not at any
definite point of space, but is loosely associated with a wide region.
 Perhaps you may think that an extended stormy area ought to
represent diffused matter in contrast to a concentrated particle. That
is not Schrödinger’s theory. The spreading is not a spreading of
density; it is an indeterminacy of position, or a wider distribution of
the probability that the particle lies within particular limits of position.
Thus if we come across Schrödinger waves uniformly filling a vessel,
the interpretation is not that the vessel is filled with matter of uniform
density, but that it contains one particle which is equally likely to be
anywhere.
The first great success of this theory was in representing the
emission of light from a hydrogen atom—a problem far outside the
scope of classical theory. The hydrogen atom consists of a proton
and electron which must be translated into their counterparts in the
sub-aether. We are not interested in what the proton is doing, so we
do not trouble about its representation by waves; what we want from
it is its field of force, that is to say, the spurious which it provides in
the equation of wave-propagation for the electron. The waves
travelling in accordance with this equation constitute Schrödinger’s
equivalent for the electron; and any solution of the equation will
correspond to some possible state of the hydrogen atom. Now it
turns out that (paying attention to the obvious physical limitation that
the waves must not anywhere be of infinite amplitude) solutions of
this wave-equation only exist for waves with particular frequencies.
Thus in a hydrogen atom the sub-aethereal waves are limited to a
particular discrete series of frequencies. Remembering that a
frequency in the sub-aether means an energy in gross experience,
the atom will accordingly have a discrete series of possible energies.
It is found that this series of energies is precisely the same as that
assigned by Bohr from his rules of quantisation (p. 191). It is a
considerable advance to have determined these energies by a wave-
theory instead of by an inexplicable mathematical rule. Further, when
applied to more complex atoms Schrödinger’s theory succeeds on
those points where the Bohr model breaks down; it always gives the
right number of energies or “orbits” to provide one orbit jump for
each observed spectral line.
 It is, however, an advantage not to pass from wave-frequency to
classical energy at this stage, but to follow the course of events in
the sub-aether a little farther. It would be difficult to think of the
electron as having two energies (i.e. being in two Bohr orbits)
simultaneously; but there is nothing to prevent waves of two different
frequencies being simultaneously present in the sub-aether. Thus
the wave-theory allows us easily to picture a condition which the
classical theory could only describe in paradoxical terms. Suppose
that two sets of waves are present. If the difference of frequency is
not very great the two systems of waves will produce “beats”. If two
broadcasting stations are transmitting on wave-lengths near together
we hear a musical note or shriek resulting from the beats of the two
carrier waves; the individual oscillations are too rapid to affect the
ear, but they combine to give beats which are slow enough to affect
the ear. In the same way the individual wave-systems in the sub-
aether are composed of oscillations too rapid to affect our gross
senses; but their beats are sometimes slow enough to come within
the octave covered by the eye. These beats are the source of the
light coming from the hydrogen atom, and mathematical calculation
shows that their frequencies are precisely those of the observed light
from hydrogen. Heterodyning of the radio carrier waves produces
sound; heterodyning of the sub-aethereal waves produces light. Not
only does this theory give the periods of the different lines in the
spectra, but it also predicts their intensities—a problem which the
older quantum theory had no means of tackling. It should, however,
be understood that the beats are not themselves to be identified with
light-waves; they are in the sub-aether, whereas light-waves are in
the aether. They provide the oscillating source which in some way
not yet traced sends out light-waves of its own period.
What precisely is the entity which we suppose to be oscillating
when we speak of the waves in the sub-aether? It is denoted by ,
and properly speaking we should regard it as an elementary
indefinable of the wave-theory. But can we give it a classical
interpretation of any kind? It seems possible to interpret it as a
probability. The probability of the particle or electron being within a
given region is proportional to the amount of in that region. So that
if is mainly concentrated in one small stormy area, it is practically
certain that the electron is there; we are then able to localise it
definitely and conceive of it as a classical particle. But the -waves
of the hydrogen atom are spread about all over the atom; and there
is no definite localisation of the electron, though some places are
more probable than others.[36]
Attention must be called to one highly important consequence of
this theory. A small enough stormy area corresponds very nearly to a
particle moving about under the classical laws of motion; it would
seem therefore that a particle definitely localised as a moving point is
strictly the limit when the stormy area is reduced to a point. But
curiously enough by continually reducing the area of the storm we
never quite reach the ideal classical particle; we approach it and
then recede from it again. We have seen that the wave-group moves
like a particle (localised somewhere within the area of the storm)
having an energy corresponding to the frequency of the waves;
therefore to imitate a particle exactly, not only must the area be
reduced to a point but the group must consist of waves of only one
frequency. The two conditions are irreconcilable. With one frequency
we can only have an infinite succession of waves not terminated by
any boundary. A boundary to the group is provided by interference of
waves of slightly different length, so that while reinforcing one
another at the centre they cancel one another at the boundary.
Roughly speaking, if the group has a diameter of 1000 wave-lengths
there must be a range of wave-length of 0.1 per cent., so that 1000
of the longest waves and 1001 of the shortest occupy the same
distance. If we take a more concentrated stormy area of diameter 10
wave-lengths the range is increased to 10 per cent.; 10 of the
longest and 11 of the shortest waves must extend the same
distance. In seeking to make the position of the particle more definite
by reducing the area we make its energy more vague by dispersing
the frequencies of the waves. So our particle can never have
simultaneously a perfectly definite position and a perfectly definite
energy; it always has a vagueness of one kind or the other
unbefitting a classical particle. Hence in delicate experiments we
must not under any circumstances expect to find particles behaving
exactly as a classical particle was supposed to do—a conclusion
which seems to be in accordance with the modern experiments on
diffraction of electrons already mentioned.
We remarked that Schrödinger’s picture of the hydrogen atom
enabled it to possess something that would be impossible on Bohr’s
theory, viz. two energies at once. For a particle or electron this is not
merely permissive, but compulsory—otherwise we can put no limits
to the region where it may be. You are not asked to imagine the state
of a particle with several energies; what is meant is that our current
picture of an electron as a particle with single energy has broken
down, and we must dive below into the sub-aether if we wish to
follow the course of events. The picture of a particle may, however,
be retained when we are not seeking high accuracy; if we do not
need to know the energy more closely than 1 per cent., a series of
energies ranging over 1 per cent, can be treated as one definite
energy.
Hitherto I have only considered the waves corresponding to one
electron; now suppose that we have a problem involving two
electrons. How shall they be represented? “Surely, that is simple
enough! We have only to take two stormy areas instead of one.” I am
afraid not. Two stormy areas would correspond to a single electron
uncertain as to which area it was located in. So long as there is the
faintest probability of the first electron being in any region, we cannot
make the Schrödinger waves there represent a probability belonging
to a second electron. Each electron wants the whole of three-
dimensional space for its waves; so Schrödinger generously allows
three dimensions for each of them. For two electrons he requires a
six-dimensional sub-aether. He then successfully applies his method
on the same lines as before. I think you will see now that
Schrödinger has given us what seemed to be a comprehensible
physical picture only to snatch it away again. His sub-aether does
not exist in physical space; it is in a “configuration space” imagined
by the mathematician for the purpose of solving his problems, and
imagined afresh with different numbers of dimensions according to
the problem proposed. It was only an accident that in the earliest
problems considered the configuration space had a close
correspondence with physical space, suggesting some degree of
objective reality of the waves. Schrödinger’s wave-mechanics is not
a physical theory but a dodge—and a very good dodge too.
The fact is that the almost universal applicability of this wave-
mechanics spoils all chance of our taking it seriously as a physical
theory. A delightful illustration of this occurs incidentally in the work
of Dirac. In one of the problems, which he solves by Schrödinger
waves, the frequency of the waves represents the number of
systems of a given kind. The wave-equation is formulated and
solved, and (just as in the problem of the hydrogen atom) it is found
that solutions only exist for a series of special values of the
frequency. Consequently the number of systems of the kind
considered must have one of a discrete series of values. In Dirac’s
problem the series turns out to be the series of integers. Accordingly
we infer that the number of systems must be either 1, 2, 3, 4, ..., but
can never be 2¾ for example. It is satisfactory that the theory should
give a result so well in accordance with our experience! But we are
not likely to be persuaded that the true explanation of why we count
in integers is afforded by a system of waves.

Principle of Indeterminacy. My apprehension lest a fourth version of

the new quantum theory should appear before the lectures were
delivered was not fulfilled; but a few months later the theory definitely
entered on a new phase. It was Heisenberg again who set in motion
the new development in the summer of 1927, and the consequences
were further elucidated by Bohr. The outcome of it is a fundamental
general principle which seems to rank in importance with the
principle of relativity. I shall here call it the “principle of
indeterminacy”.
The gist of it can be stated as follows: a particle may have
position or it may have velocity but it cannot in any exact sense have
both.
If we are content with a certain margin of inaccuracy and if we
are content with statements that claim no certainty but only high
probability, then it is possible to ascribe both position and velocity to
a particle. But if we strive after a more accurate specification of
position a very remarkable thing happens; the greater accuracy can
be attained, but it is compensated by a greater inaccuracy in the
specification of the velocity. Similarly if the specification of the
velocity is made more accurate the position becomes less
determinate.
Suppose for example that we wish to know the position and
velocity of an electron at a given moment. Theoretically it would be
possible to fix the position with a probable error of about ¹⁄₁₀₀₀ of a
millimetre and the velocity with a probable error of 1 kilometre per
second. But an error of ¹⁄₁₀₀₀ of a millimetre is large compared with
that of some of our space measurements; is there no conceivable
way of fixing the position to ¹⁄₁₀₀₀₀ of a millimetre? Certainly; but in
that case it will only be possible to fix the velocity with an error of 10
kilometres per second.
The conditions of our exploration of the secrets of Nature are
such that the more we bring to light the secret of position the more
the secret of velocity is hidden. They are like the old man and
woman in the weather-glass; as one comes out of one door, the
other retires behind the other door. When we encounter unexpected
obstacles in finding out something which we wish to know, there are
two possible courses to take. It may be that the right course is to
treat the obstacle as a spur to further efforts; but there is a second
possibility—that we have been trying to find something which does
not exist. You will remember that that was how the relativity theory
accounted for the apparent concealment of our velocity through the
aether.
When the concealment is found to be perfectly systematic, then
we must banish the corresponding entity from the physical world.
There is really no option. The link with our consciousness is
completely broken. When we cannot point to any causal effect on
anything that comes into our experience, the entity merely becomes
part of the unknown—undifferentiated from the rest of the vast
unknown. From time to time physical discoveries are made; and new
entities, coming out of the unknown, become connected to our
experience and are duly named. But to leave a lot of unattached
labels floating in the as yet undifferentiated unknown in the hope that
they may come in useful later on, is no particular sign of prescience
and is not helpful to science. From this point of view we assert that
the description of the position and velocity of an electron beyond a
limited number of places of decimals is an attempt to describe
something that does not exist; although curiously enough the
description of position or of velocity if it had stood alone might have
been allowable.
Ever since Einstein’s theory showed the importance of securing
that the physical quantities which we talk about are actually
connected to our experience, we have been on our guard to some
extent against meaningless terms. Thus distance is defined by
certain operations of measurement and not with reference to
nonsensical conceptions such as the “amount of emptiness”
between two points. The minute distances referred to in atomic
physics naturally aroused some suspicion, since it is not always easy
to say how the postulated measurements could be imagined to be
carried out. I would not like to assert that this point has been cleared
up; but at any rate it did not seem possible to make a clean sweep of
all minute distances, because cases could be cited in which there
seemed no natural limit to the accuracy of determination of position.
Similarly there are ways of determining momentum apparently
unlimited in accuracy. What escaped notice was that the two
measurements interfere with one another in a systematic way, so
that the combination of position with momentum, legitimate on the
large scale, becomes indefinable on the small scale. The principle of
indeterminacy is scientifically stated as follows: if is a co-ordinate
and the corresponding momentum, the necessary uncertainty of
our knowledge of multiplied by the uncertainty of is of the order
of magnitude of the quantum constant .
A general kind of reason for this can be seen without much
difficulty. Suppose it is a question of knowing the position and
momentum of an electron. So long as the electron is not interacting
with the rest of the universe we cannot be aware of it. We must take
our chance of obtaining knowledge of it at moments when it is
interacting with something and thereby producing effects that can be
observed. But in any such interaction a complete quantum is
involved; and the passage of this quantum, altering to an important
extent the conditions at the moment of our observation, makes the
information out of date even as we obtain it.
Suppose that (ideally) an electron is observed under a powerful
microscope in order to determine its position with great accuracy. For
it to be seen at all it must be illuminated and scatter light to reach the
eye. The least it can scatter is one quantum. In scattering this it
receives from the light a kick of unpredictable amount; we can only
state the respective probabilities of kicks of different amounts. Thus
the condition of our ascertaining the position is that we disturb the
electron in an incalculable way which will prevent our subsequently
ascertaining how much momentum it had. However, we shall be able
to ascertain the momentum with an uncertainty represented by the
kick, and if the probable kick is small the probable error will be small.
To keep the kick small we must use a quantum of small energy, that
is to say, light of long wave-length. But to use long wave-length
reduces the accuracy of our microscope. The longer the waves, the
larger the diffraction images. And it must be remembered that it
takes a great many quanta to outline the diffraction image; our one
scattered quantum can only stimulate one atom in the retina of the
eye, at some haphazard point within the theoretical diffraction image.
Thus there will be an uncertainty in our determination of position of
the electron proportional to the size of the diffraction image. We are
in a dilemma. We can improve the determination of the position with
the microscope by using light of shorter wave-length, but that gives
the electron a greater kick and spoils the subsequent determination
of momentum.
A picturesque illustration of the same dilemma is afforded if we
imagine ourselves trying to see one of the electrons in an atom. For
such finicking work it is no use employing ordinary light to see with; it
is far too gross, its wave-length being greater than the whole atom.
We must use fine-grained illumination and train our eyes to see with
radiation of short wave-length—with X-rays in fact. It is well to
remember that X-rays have a rather disastrous effect on atoms, so
we had better use them sparingly. The least amount we can use is