Tut Sheet 4 SCH RNT Freeb SHO
Tut Sheet 4 SCH RNT Freeb SHO
Tut Sheet 4 SCH RNT Freeb SHO
Tutorial Sheet 4
(Wave Mechanics)
P69: A beam of electrons with energy E = 4 eV approaches from left hand side a
potential barrier defined as V(x)=0 for x<0 and V(x)=5 eV for x>0. Find the value of
x inside the barrier for which the probability density is one-fourth the probability
density at x=0.
P70*: A beam of particles with energy E approaches from left hand side, a potential
barrier defined by V=0 for x<0 and V=Vo for x>0, where Vo>E.
(a) Find the value of x=xo (xo>0), for which the probability density is 1/e times the
probability density at x=0.
(b) Take maximum allowed uncertainty x for the particle to be localized in the
classically forbidden region as xo. Find the uncertainty this would cause in the
energy of the particle. Can then one be sure that its energy E is less than Vo.
P72*: A beam of particles of energy E and de Broglie wavelength , traveling along the
positive x-axis in potential free region, encounters a one-dimensional potential
barrier of height V=E and width L.
(a) Obtain an expression for the transmission coefficient.
(b) For what value of L (in terms of ), will the reflection coefficient be half?
P73: A beam of particles (mass 500 keV/c2) moving along negative x direction of energy
3 eV is incident on a potential given by
V= for x0
V= 5 eV for 0<xa
V= 0 for x>a
The value of a is equal to the de-Broglie wavelength of the particles (in the region
x>a). Write the wave function of the particles in terms of the amplitude of the
incident wave. What is the ratio of the probabilities of finding the particle at x = a/2
and at x = a.
4
P74: A beam of particle of mass m and energy Vo (where Vo is positive constant) is
3
incident from left on the following potential barrier.
V = 0 for x < a and for x > + a
V = Vo for a x + a
3
Where a . Write the wave functions in all the three regions and apply
2mVo
boundary conditions, clearly stating them. Find the transmission coefficient of the
particles.
V=0
x=0 x=d
V2
V1
-L 0 L
x
P78*: A particle of mass m is confined to a one-dimensional box described by V=0 for
0<x<L and for 2L<x<3L; V=Vo for L<x<2L and V= , everywhere else. It is given
that the ground state wave function of the particle is independent of x between
L<x<2L
Vo
0 L 2L 3L
x
P79: Consider a one-dimensional potential shown in the figure below. This is described
by V= for x0; V=-Vo for 0<x<L and V=0 for xL.
(a) For a beam of particle coming from right with energy E>0, what is the reflection
coefficient? Can you guess the answer without doing the calculation?
(b) Find the equation that governs the energy for the bound state. Normalize the wave
function for this case.
(c) For a given value of L, what should be the value of Vo, so that there is only one
bound state.
-Vo
0 L
x
V=V0 for x < 0; V=0 for 0 < x < L; V=V0 for x> L
(a) Find the value of V0, for which the particle will have only one bound state at
E=V0/2.
(b) Find the values of x in the classically forbidden region, for which the probability of
finding the particle will be (1/e) of the value at x = L.
P82: Consider the following one dimensional potential well (Vo is positive)
V 0 for x 0
V Vo for 0 x d
V 4Vo for x d
Find the un-normalized wave function for a particle bound in the well and roughly
plot it in your answer-book for the ground state. Find the equation that determines
the energy of the quantized levels. Find the least value of d for which a bound
state will exist.
P83*: A particle of mass m is in the second excited state (n=3) of an 1-d infinite square
well which extends from x=0 to x=L. Suddenly the well expands to double its size
(i.e. from x=0 to x=2L), leaving the wave function undisturbed momentarily. If the
energy of the particle is now measured, what are the probabilities of finding it in the
ground state and in the first excited state (n=2)? At a later time the wave function of
the particle is given by the same functional form, but now extends between x=0 and
2L. Find out the probability of finding the particle in the ground state and n=2 state
now.
M: Harmonic Oscillator:
1
P86: A particle of mass m is under the influence of a potential V m 2 x 2 , where x is
2
the displacement from origin and is a constant. The ground state wave function of
m 2
x
this particle is given by (x) Ae 2
. Find the mean values x , x 2 , px
and px 2 corresponding to this wave function and the uncertainty product. For
integer k, you may use the following integrals.
1 3 ...(2k-1)
0 u e du
2 k bu2
2k 1 b 2 k 1
k!
u
2 k 1 bu2
e du
0
2bk 1
P87: A particle of mass m is under the influence of a potential V= m2x2, where x is
m 2
x
the displacement from origin and is a constant. Show that ( x) Axe 2
is a
solution of the one-dimensional Schrodinger equation with the above potential. Find
the value of energy for this state.
P88*: For a particle bound in a particular one-dimensional potential with a property that
V(0)=0, the two solutions of Schrdinger Equation are given by (x) and
1
x 1(x). The energies corresponding to these solutions are E1 and E2 where E1 E2.
Find the un-normalized (x) , the ratio E2/E1 and V(x).
1
P89*: (a) One of the solutions of the Schrdinger equation for a particle of mass m
1
experiencing a potential V kx 2 , k being constant is Axe x . Find out the value
2
2
of and the energy.
(b) Find the value of x xo , beyond which the region is classically forbidden for the
particle.
(c) Consider a particle bound between xo x xo in the following potential
V Vo for x xo
1 2
V kx for xo x xo
2
V Vo for x xo
Can Axe x still be a valid wave function for the region xo x xo ? Please
2
1 d 2 dR 2m
r E
e2 1
R 0
r 2 dr dr 2 4 or r2
where the symbols have their usual meaning. For the ground state of hydrogen atom
=0. For this state
r
(a) Show that (r , t 0) Ae a is a solution of this equation. Find the values of
A, a and the ground state energy.
(b) Calculate the mean distance, root mean square distance and the most probable
distance between the electron and the nucleus in terms of the Bohr radius.
(c) What are the classical and quantum mechanical probabilities of finding the
electron ar r>2a.
x
p
e x dx (p 1) p(p); p 1 0
0
P91: A particle is bound in a potential well of the type V(r) =-Vo for r <a and V(r)=0 for
r>a. The wave function (r , , , t 0) for this case is written as a product of three
functions as (r , , , t 0) R(r ) ( ) () and the radial part of the
Schrdinger equation can be written as follows for =0.
1 d 2 dR 2m
r E V R 0
r 2 dr dr 2
The symbols have their usual meaning. Solve this equation to find the equation
which would lead to energy states.
(Hint: Use the substitution (r)=R(r) r, to reduce the equation into a more familiar
differential equation)
P92*: The and parts of the Schrdinger equation are given as follows.
sin d d 2
sin ( 1)sin ml
2
d d
1 d 2
ml 2
d 2
If we substitute w = cos and write P for in the first equation, we get the
following equation.
d P dP ml 2
1- w dw
2
2
2w ( 1) P 0
2
dw 1 w2
It is given that P Aw 2 B is a solution of this equation. Find the possible values
A
of , and m . Write the form of and for these cases.
B