Chapter_4.pptx

1. Department of Physics - MIT, Manipal 1
Chapter 4
QUANTUM MECHANICS
• OBJECTIVES:
• To learn the application of Schrödinger equation to a bound particle and to
learn the quantized nature of the bound particle, its expectation values and
physical significance.
• To understand the tunneling behavior of a particle incident on a potential
barrier.
• To understand the behavior of quantum oscillator.

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An Interpretation of Quantum Mechanics
For an electromagnetic radiation, the probability of finding a photon per unit volume
is related to the amplitude E of the electric field as
2
Y
PROBABILIT
E
V

Similarly
 = wave function of a particle
= amplitude of the de Broglie wave
= probability amplitude
Time dependent wave function for a system:
(rj,t) = (rj) e–it
rj is the position vector of the jTH particle in the system.

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 contains all the information about the particle
 → imaginary entity
→ no physical significance
PROBABILITY DENSITY = ||
2
||
2
→ real and positive
→ relative probability per unit volume that the particle will be found at any
given point in the volume
dx
P 2
b
a
ab 
 
= area under the probability density
curve from a to b.
Normalization condition:
1
dx 
∫
∞
∞
-
2


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Mathematical features of a physically reasonable wave function (x) for a
system:
(i) (x) may be a complex function or a real function, depending on the system;
(ii) (x), must be finite, continuous and single valued everywhere;
(iii) The space derivatives of , must be finite, continuous and single valued
everywhere;
(iv)  must be normalizable.

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dx
x
x 

 




• The expectation value of any function f(x) associated with the particle is
dx
)
x
(
f
)
x
(
f 

 




• Measurable quantities of the particle (energy, momentum, etc) can be derived from 
• x = expectation value of x (i.e. the average position at which one expects to find the
particle after many measurements)
Time independent Schrödinger equation
for a particle of mass m confined to moving along x axis and interacting with its
environment through a potential energy function U(x) and E = total energy of the
system (particle and its environment)





 E
U
dx
d
m
2 2
2
2


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Particle in an Infinite Potential Well (Particle in a “Box”)
U(x) = 0, for 0 < x < L,
U (x) =  , for x < 0, x > L
U (x) =  , for x < 0, x > L [here (x) = 0 ]
In 0 < x < L, U = 0, the Schrödinger
equation is
2
2 2
2
0
d m
E
dx


 
2m E
k 
2
2
2
where
d
k ,
dx

  
(x) = A sin(kx) + B cos(kx) where A and B are constants
At x = 0,  = 0 So, 0 = A sin 0 + B cos 0 or B = 0 ,
At x = L ,  = 0 ,
0 = A sin(kL) + B cos(kL) = A sin(kL) + 0 ,

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since A  0 , sin(kL) = 0 .
 k L = n π; ( n = 1, 2, 3, ………..)











2
λ
n
L
or
L
n
λ
2
k




 n
L
E
m
2
k L
,
E
m
2
k


Each value of the integer n corresponds to a quantized energy value, En , where
2
2
2
n n
L
m
8
h
E 







 n = 1, 2 , 3 …..
2
2
1
L
m
8
h
E 
The lowest allowed energy (n = 1),
E1 is the ground state energy for the particle in a box
Excited states → n = 2, 3, 4, . . . .
Energies: En → 4E1 , 9E1 , 16E1 , . . . . .

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To find the constant A, apply normalization condition
 















 


L
0
2
2
2
1
dx
L
x
n
sin
A
or
1
dx
 











 

L
0
2
1
dx
L
x
n
2
cos
1
A 2
1
)
( L
x
n
L
2 sin
)
x
(
n



Solving we get
)
( L
x
n
L
2 2
sin
)
x
(
Pn


WAVE FUNCTION
PROBABILITY DENSITY
L
2
A 

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Sketch of (a) wave function, (b) Probability density for a particle in potential
well of infinite height

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A Particle in a Potential Well of Finite Height
A particle is trapped in the well. The total energy E
of the particle-well system is less than U
U(x) = 0 , 0 < x < L,
U (x) = U , x < 0, x > L
• Particle energy E < U ; classically the particle is
permanently bound in the potential well.
• However, according to quantum mechanics, a finite probability exists that the
particle can be found outside the well even if E < U.
• The Schrödinger equation outside the finite well in regions I and III is:





 2
2
2
2
C
)
E
U
(
m
2
dx
d

)
E
U
(
m
2
C
where 2
2




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General solution of the above equation is
x
C
x
C
e
B
e
A
)
x
( 



In region I, B = 0;

I
= A eC x for x < 0
In region III, A = 0;

III
= B e–C x for x > L
In region II,
ћ2
dx2
d2
II
+
2m
E 
II
= 0
k2
II = F sin kx + G cos kx

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0
x
0
x dx
dx
d
)
(
)
( 0
0




II
I
II
I
dψ
ψ
ψ
ψ
L
x
L
x dx
d
dx
d
)
(
)
( L
L




III
II
III
II
ψ
ψ
ψ
ψ
PROBABILITY DENSITIES
WAVE FUNCTIONS
A, B, F, G values can be obtained by applying boundary conditions.

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Tunneling Through a Potential Energy Barrier
• Consider a particle of energy E
approaching a potential barrier of height
U, (E<U).
• Since E<U, classically the regions II and
III shown in the figure are forbidden to
the particle incident from left.
• But according to quantum mechanics, all regions are accessible to the particle,
regardless of its energy.
• An approximate expression for the transmission coefficient, when T<< 1 is
T ≈ e−2CL , where 𝐶 =
2 𝑚 𝑈−𝐸
ℏ
• Since the particles must be either reflected or transmitted:
R + T =1

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The Simple Harmonic Oscillator
• Consider a particle that is subject to a linear restoring force 𝐹 = −𝑘𝑥, where k is a
constant and x is the position of the particle relative to equilibrium (at equilibrium
position x=0).
• Classically, the potential energy of the system is,
𝑈 =
1
2
𝑘𝑥2 =
1
2
𝑚𝜔2𝑥2
where the angular frequency of vibration is 𝜔 = 𝑘/𝑚.
• The total energy E of the system is,
𝐸 = 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝐾 + 𝑈 =
1
2
𝑘𝐴2
=
1
2
𝑚𝜔2
𝐴2
where A is the amplitude of motion.

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• A quantum mechanical model for simple harmonic oscillator can be obtained by
substituting 𝑈 =
1
2
𝑚𝜔2𝑥2 in Schrödinger equation:
−
ℏ2
2𝑚
𝑑2
𝑑𝑥2 +
1
2
𝑚𝜔2𝑥2 = 𝐸
• The solution for the above equation is
 = 𝐵𝑒−𝐶𝑥2
where 𝐶 = 𝑚𝜔/2ℏ and 𝐸 =
1
2
ℏ𝜔.
• Energy of a state is given by
𝐸𝑛 = 𝑛 +
1
2
ℏ𝜔; 𝑛 = 0, 1, 2 … .
• The state n=0 corresponds to the ground
state, whose energy is 𝐸0 =
1
2
ℏ𝜔