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Wavefunction
• Quantum mechanics acknowledges the waveparticle duality of matter by supposing that,
rather than traveling along a definite path, a
particle is distributed through space like a
wave. The wave that in quantum mechanics
replaces the classical concept of particle
trajectory is called a wavefunction, ψ (“psi”).
A wave function in quantum
mechanics describes the quantum
state of an isolated system of one or
more particles. There is one wave
function
containing
all
the
information about the entire system,
not a separate wave function for
each particle in the system.
Wave equation for the harmonic motion
d 2 ( x)
 2 



  ( x)
2
dx
  
2
 2
  2
 4
 d 2 ( x)

  ( x)
2
dx

1 2
p2
E  mv  V 
V
2
2m
p  [2mE  V ]

h

p
1
2
h
[2mE  V ]
1
2
 h 2  d 2 ( x)
  2 
 ( E  V ) ( x)
2
8

m
dx


Postulates of Quantum Mechanics
Postulate 1:
State and wave functions. Born interpretation
The state of a quantum mechanical system is completely specified by a wave
function ψ (r,t) that depends on the coordinates of the particles (r) and time t. These
functions are called wave functions or state functions.
For 2 particle system:
  ( x1 , y1 , z1 , x2 , y2 , z2 , t )
Wave function contains all the information about a system.
wave function
 classical trajectory
(Quantum mechanics)
(Newtonian mechanics)
Meaning of wave function:
P(r) = |ψ|2 =  d
=> the probability that the particle can be found at a particular point x and a
particular time t. (Born’s / Copenhagen interpretation)
*
Implications of Born’s Interpretation
(1) Positivity:
P(r) >= 0
The sign of a wavefunction has no direct physical significance:
The positive and negative regions of this wavefunction both
correspond to the same probability distribution.
(2) Normalization:
*

  d  1
all _ space
i.e. the probability of finding the particle in the universe is 1.
Physically acceptable wave function
 The wave function and its first derivative must
be:
1) Finite. The wave function must be single
valued. This means that for any given values
of x and t , Ψ(x,t) must have a unique value.
This is a way of guaranteeing that there is
only a single value for the probability of the
system being in a given state.
2. Square-integrable
The wave function must be square-integrable. In other
words, the integral of |Ψ|2 over all space must be
finite. This is another way of saying that it must be
possible to use |Ψ|2 as a probability density, since any
probability density must integrate over all space to give
a value of 1, which is clearly not possible if the integral
of |Ψ|2 is infinite. One consequence of this proposal is
that must tend to 0 for infinite distances.
Continuous wavefunction
• A rapid change would mean that the
derivative of the function was very large
(either a very large positive or negative
number). In the limit of a step function, this
would imply an infinite derivative. Since the
momentum of the system is found using the
momentum operator, which is a first order
derivative, this would imply an infinite
momentum, which is not possible in a
physically realistic system.
Continuous First derivative
1. All first-order derivatives of the wave
function must be continuous. Following the
same reasoning as in condition 3, a
discontinuous first derivative would imply an
infinite second derivative, and since the
energy of the system is found using the
second derivative, a discontinuous first
derivative would imply an infinite energy,
which again is not physically realistic.
Acceptable or Not ??
Acceptable or not acceptable ??
(i )e  x (0,  )
x
(ii )e (,  )
sin x
(iii )
x
(iv ) sin
1
x
Exp(x)
Sinx/x
Sin-1x
Postulate 2
To every physical property, observable in
classical mechanics, there corresponds
a linear, hermitian operator in quantum
mechanics.
Operator
• A rule that transforms a given function into
another function
Operator
• Example. Apply the following operators on the
given functions:
• (a) Operator d/dx and function x2.
• (b) Operator d2/dx2 and function 4x2.
• (c) Operator (∂/∂y)x and function xy2.
• (d) Operator −iћd/dx and function exp(−ikx).
• (e) Operator −ћ2d2/dx2 and function exp(−ikx).
Identifying the operators
Hermitian Operator
•
Hermitian operators have two properties
that forms the basis of quantum mechanics
(i) Eigen value of a Hermitian operator are real.
(ii) Eigenfunctions of Hermitian operators are
orthogonal to each other or can be made
orthogonal by taking linear combinations of
them.
Hermitian operator
ˆ satisfies
A hermitian operator A
*
*
f
Â
g
dx

g(
Â
f)
dx ; if f and g are well behaved


• Prove Operator x is Hermitian.
xrs 
*
 r x s dx

  rx

* *
 s dx
*
  ( s* x * r )* dx  x *sr
Hermitian operator or not ??

(i )
x

(ii )  i
x
2
(iii ) 2
x
Linear Operator
• A linear operator has the following properties



A f1  f 2   A f1  A f 2


Acf   c A f
Linear operator
Derivative
integrals
log
√
Normalized wave function:
N 2  * dx  1
Orthogonal wave functions:
Orthonormal set wave functions:



m*n dx   mn  1, if m  n
 0, if m  n
Hermitian operator
• Example. Prove that the momentum operator
(in one dimension) is Hermitian.


 *
* d
( p x ) rs   r  s dx    r s
i  dx
i




d * 
  s  r dx 
dx



Postulate 3
In any measurement of the observable associated
with the operator Â, the only values that will ever be
observed are the eigenvalues ‘a’ which satisfy the
eigenvalue equation:
This is the postulate that the values of dynamical
variables are quantized in quantum mechanics.
Eigen Function and Eigen value

A f ( x)  kf ( x)

f(x) is eigenfunct ion of A with eigen valu e k
Q: What are the eigenfunctions and eigenvalues
of the operator d/dx ?
Eigen function and eigen value
f x   e
ikx
Is it eigen function of momentum operator ?
What is eigen value ?
Eigenvalue equation
Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
^
 
Example: eikx is an eigenfunction of a operator Px = -ih
x
F(x) = eikx
= -i h  eikx
x
= -i2 hk2eikx
= h k2eikx
Thus eikx is an eigenfunction
Significance of commutation rules
• The eigenvalue of a Hermitian operator is real.
• A real eigenvalue means that the physical quantity for which the operator
stands for can be measured experimentally.
• The eigenvalues of two commuting operators can be computed by using
the common set of eigenfunctions.
If the two operators commute, then it is possible to measure the
simultaneously the precise value of both the physical quantities for
which the operators stand for.
Question: Find commutator of the operators x and px
Is it expected to be a non-zero or zero quantity?
Hint: Heisenberg Uncertainty Principle
Commute or not ??
• Operator x and d/dx
They don’t



d
dx
Ax
B
 
A B f ( x)  xf , ( x)
 
B A f ( x) 
d
[ xf ( x)]  xf , ( x)  f ( x)
dx
  
A, B f ( x)   f ( x)
  
A, B  1
Postulate 4
For a system in a state described by a
normalized wave function , the average or
expectation value of the observable
corresponding to A is given by:
Mean value theorem
Expectation value in general:
The fourth postulates states what will be measured when large number
of identical systems are interrogated one time. Only after large number
of measurements will it converge to <a>.
In QM, the act of the measurement causes the system to “collapse” into
a single eigenstate and in the absence of an external perturbation it will
remain in that eigenstate.
Postulate 5
The wave function of a system evolves in time in
accordance with the time dependent
Schrodinger equation:
Schrodinger Equation
Time independent Schrodinger equation
General form:
H = E 
E= T + V
Schrödinger Representation – Schrödinger Equation
Time dependent Schrödinger Equation
i
 ( x , y , z , t )
 H ( x , y , z , t )( x , y , z , t )
t
Developed through analogy to Maxwell’s equations and knowledge of
the Bohr model of the H atom.
H classical
Hamiltonian
Q.M.
p2

V
2m
Sum of kinetic energy and potential energy.
kinetic potential
energy energy
 2 2
H
 V (x)
2
2m  x
 2 2
H
  V ( x, y, z )
2m
p  i

x
one dimension
recall
three dimensions
2
2
2
 


2
2
 x  y  z2
2
The potential, V, makes one problem different form another H atom, harmonic oscillator.
Copyright – Michael D. Fayer, 2007
Getting the Time Independent Schrödinger Equation
( x , y, z , t )
wavefunction

i
( x , y, z , t )  H ( x , y, z , t )( x, y, z, t )
t
If the energy is independent of time
H ( x, y, z )
Try solution
( x , y , z , t )   ( x , y , z )F ( t )
product of spatial function and time function
Then

i
 ( x , y, z )F ( t )  H ( x , y, z ) ( x , y, z )F ( t )
t

i  ( x, y, z )
F ( t )  F ( t ) H ( x , y, z ) ( x , y, z )
t
independent of t
divide through by
  F
independent of x, y, z
Copyright – Michael D. Fayer, 2007
i
dF ( t )
dt  H ( x , y, z )  ( x , y, z )
F (t )
 ( x, y, z )
depends only
on t
depends only
on x, y, z
Can only be true for any x, y, z, t if both sides equal a constant.
Changing t on the left doesn’t change the value on the right.
Changing x, y, z on right doesn’t change value on left.
Equal constant
i
dF
dt  E  H 
F

Copyright – Michael D. Fayer, 2007
i
dF
dt  E  H 
F

Both sides equal a constant, E.
H ( x, y, z ) ( x, y, z )  E  ( x, y, z )
Energy eigenvalue problem – time independent Schrödinger Equation
H is energy operator.
Operate on  get  back times a number.
’s are energy eigenkets; eigenfunctions; wavefunctions.
E
Energy Eigenvalues
Observable values of energy
Copyright – Michael D. Fayer, 2007
Time Dependent Equation (H time independent)
i
i
dF ( t )
dt  E
F (t )
dF ( t )
 E F (t )
dt
dF ( t )
i
  E dt .
F (t )
ln F  
i Et
Integrate both sides
C
F (t )  e  i E t /  e  i t
Time dependent part of wavefunction for time independent Hamiltonian.
Time dependent phase factor used in wave packet problem.
Copyright – Michael D. Fayer, 2007