199 Unit-1 Quantum Mechanics
199 Unit-1 Quantum Mechanics
199 Unit-1 Quantum Mechanics
Unit-1
Quantum mechanics and Quantum Computing
INTRODUCTION:
In 19th century all the scientists believed that NEWTONS LAWS & E.M LAWS
are the basic laws of physics and they can explain all the laws of physics. The
branch of physics that can be explained basing on NEWTONIAN MECHANICS is
called CLASICAL MECHANICS. Classical mechanics failed to explain the concepts
1. Hydrogen spectrum
2. Blackbody radiation
3. Specific heat
4. Stability atom
5. Motion of micro particles
h h m0 h
But λ = = =
m0 v m0 2eV 2eVm 0
h
⇒ λ=
2eVm 0
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Using the values h=6.625X10 -34
Js, m0=9.1X10 Kg and e=1.6X10 C, we get
-31 -19
12 .26 0
λ= A
V
This expression is for non-relativistic case since relative variation of mass with
velocity is not considered. Thus accelerated electrons exhibit wave nature
corresponding to X-ray wavelength.
MATTER WAVES:
Def: The waves which are generated due to the motion of material particles are
called matter waves.
PROPERTIES OF MATTER WAVES:
h h
The wavelength of matter waves is given by λ = =
mv P
Lighter is the particle greater will be the wavelength associated with it.
Smaller is the velocity of particle, greater will be the wavelength associated with
it.
For v=0, λ = α this shows that matter waves are generated only by the
motion of
particles.
ω
Phase velocity of matter waves is given by vph =
k
These waves are produced whether the particles are charged are neutral.
These
waves are not electro magnetic waves.
The velocity of matter waves depend up on the velocity of material particle.
The velocity of material particle is greater than the velocity of light.
The wave and particle properties never appear together.
The wave nature of matter introduces an uncertainty in the location of position
of
the particle.
HEISENBERG UNCERTAINTY PRINCIPLE:-
Heisenberg proposed this uncertainty principle in the year 1927.
If a particle is moving according to classical mechanics, we can find its position
and momentum
In wave mechanics, we regard a moving particle as a wave group. This can be
understand by considering the following wave groups.
Consider
narrow wave
group:
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In this case higher will be the accuracy of locating the particle at the same
time
one cannot define the wavelength (λ) accurately since the measurement of
particle’s momentum also less accurate.
Consider a wide wave group here λ can be well-defined. Hence the
measurement
of momentum becomes more accurate.
Principle: It is impossible to know both the exact position and exact momentum
of an object at the same time.
If ∆x and ∆p are the uncertainties in the position and momentum
h
respectively then according to the principle ∆x ∆p≥
4π
h
using = we can write the above as ∆x ∆p≥
2π 2
SCHRÖDINGER’S TIME INDEPENDENT WAVE EQUATION:-
Schrödinger’s equation is a D.E for debroglie’s waves associated with a
particle
and thus describes the motion of the particle.
Now we introduce a wave function ψ associated with the moving particle.
It is the basic equation in Quantum mechanics. It is derived from the de
Broglie’s
concept of matter waves.
According to deBroglie’s a moving particle of mass ‘m’ is associated with a
wave
whose wavelength is λ.
In classical mechanics the equation for a plane wave moving along X- direction
is
2π
given by y = a sin (x-vt) ------- (1)
λ
Where y = displacement
a = amplitude
x = position coordinate in x direction
Differentiating the above equation w.r.t ‘x’ we get
dy 2π 2π
=a cos (x-vt)
dx λ λ
2π
⇒ d 2y = - 4π2 a sin
2 2
(x-vt)
dx λ λ
dx λ
dx λ
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Schrodinger derived the time independent wave equation in Quantum mechanics
h
by replacing y with ψ and λ =
mv
d 2ψ 4π 2 m 2 v 2 ψ
+ = 0 ------- (3)
dx 2 h2
We know that the total energy is sum of Kinetic energy and Potential energy.
Total energy E = K.E(U)+P.E(V)
1
E = U+V ⇒ U=E-V ⇒ mv2 = E-V
2
⇒ m2v2 = 2m(E-V)
d 2ψ 4π 2
Substituting this value in equation(3), we get + 2 2m(E-V)ψ = 0
dx 2 h
⇒dψ 8π 2
2
2
+ 2
m(E-V)ψ = 0
dx h
⇒dψ
2
2m
2
+ 2 (E-V)ψ =
dx
0
For the same particle moving in 3 dimensional space, the equation becomes
2m
∇2 ψ + (E-V)
2
ψ= 0
∂2 ∂2 ∂2
where ∇is known as Laplacian operator. And ∇2 = + +
∂x 2 ∂y 2 ∂z 2
For a free particle, V = 0, then Schrdinger’s equation for a particle is
2m ψ
∇2 ψ + 2 E
=0
♣ ψ has no physical meaning but when we multiply this with its complex
conjugate
the product ψ2 has physical meaning.
= � ⇒ = ⇒ =E �-------------------(3)
♣ The schrodienger time independent wave equation becomes
2m
+ (E-V)ψ = 0
2
2m 2m
⇒ + 2 Eψ - 2 Vψ = 0
2m Eψ + Vψ
⇒ =
2
⇒ Eψ = + Vψ ------------- (4)
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