Download 6. Quantum Mechanics II

Review for Exam 2
The Schrodinger Eqn.
What is important?
The Schrödinger Wave Equation
Operators, expectation values
The simple harmonic oscillator
Quantum mechanics applied to the
Hydrogen atom: quantum number,
energy and angular momentum
Study Chapters 5 and 7 hard!
Erwin Schrödinger (1887-1961)
The Schrödinger Wave Equation
The Schrödinger Wave Equation for the wave function (x,t) for a
particle in a potential V(x,t) in one dimension is:
2

2
i

V 
2
t
2m x
where
i  1
The Schrodinger Equation is the fundamental equation of
Quantum Mechanics.
Note that it’s very different from the classical wave equation.
But, except for its inherent complexity (the i), it will have similar
solutions.
Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time: V = V(x).
The Schrödinger equation’s dependence on time and position can then
be separated. Let:
 ( x, t )   ( x) f (t )
2

2
And substitute into: i

V 
2
t
2m x
which yields:
f (t )
 2 f (t )  2 ( x)
i ( x)

 V ( x) ( x) f (t )
2
t
2m
x
Now divide by (x) f(t):
1 df (t )
 2 1  2 ( x)
i

 V ( x)
2
f (t ) t
2m  ( x) x
The left side depends only on t, and the right side
depends only on x. So each side must be equal to
a constant. The time-dependent side is:
i
1 df
B
f t
1 df
i
B
f t
Time-Independent Schrödinger
Wave Equation
f
Multiply both sides by f /iħ:
t
 B f /i
which is an easy differential
equation to solve:
f (t )  e Bt / i  eiBt /
But recall our solution for the free particle:  ( x, t )  e
i  kx wt 
in which f(t) = exp(-iwt), so: w = B / ħ or B = ħw, which means that: B = E !
f (t )  eiEt /
So multiplying by (x), the spatial Schrödinger equation becomes:
d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx
2
Stationary States
The wave function can now be written as:
( x, t )   ( x)eiEt /   ( x)eiwt
The probability density becomes:
*   * ( x) eiwt  ( x) eiwt
  ( x)
2
The probability distribution is constant in time.
This is a standing-wave phenomenon and is called a stationary state.
Most important quantum-mechanical problems will have stationary-state
solutions. Always look for them first.
Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is
given in the equation
P( x)dx   ( x, t )( x, t )dx
The probability of the particle being between x1 and x2 is given by
x2
P    dx
x1
The wave function must also be normalized so that the probability
of the particle being somewhere on the x axis is 1.



 ( x, t )( x, t )dx  1
Expectation Values
In quantum mechanics, we’ll compute expectation values.
The expectation value, x , is the weighted average of a
given quantity. In general, the expected value of x is:
x  P1 x1  P2 x2 
 PN xN 
P x
i
i
i
If there are an infinite number of possibilities, and x is continuous:

x  P( x) x dx
Quantum-mechanically:
x 



 ( x) x dx   ( x)  * ( x) x dx   * ( x) x  ( x) dx
2
And the expectation of some function of x, g(x):
g ( x) 

 * ( x) g ( x)  ( x) dx
Bra-Ket Notation
This expression is so important that physicists have a special
notation for it.

g ( x)   * ( x) g ( x)  ( x) dx 
The entire expression is called a bracket.
And  | is called the bra with |  the ket.
The normalization condition is then:
|   1
|g|
General Solution of the Schrödinger
Wave Equation when V = 0
In free space (with V = 0), the wave function is:
( x, t )  Aei ( kxwt )  A[cos(kx  wt )  i sin(kx  wt )]
which is a sine wave moving in the x direction.
Notice that, unlike classical waves, we are not taking the real part
of this function.  is, in fact, complex.
In general, the wave function is complex.
But the physically measurable quantities must be real.
These include the probability, position, momentum, and energy.
Momentum Operator
To find the operator for p, consider the derivative of the wave function
of a free particle with respect to x:

 i ( kx wt )

[e
]  ikei ( kx wt )  ik
x x

 p
With k = p / ħ we have:
 i 
x
 
This yields:

p   i
x
This suggests we define the momentum operator as: pˆ  i
The expectation value of the momentum is:

p  i   * ( x, t )

 ( x, t )
dx
x

.
x
Position and Energy Operators
The position x is its own operator. Done.
Energy operator: Note that the time derivative of the free-particle
wave function is:
  i ( kx wt )
 [e
]  iwei ( kx wt )  iw
t
t
Substituting w  E / ħ yields: E   i

t
This suggests defining the energy operator as:

Eˆ  i
t
The expectation value of the energy is:
E i
( x, t )
 ( x, t )
dx
t



*
Simple Harmonic Oscillator
Simple harmonic
oscillators describe
many physical
situations: springs,
diatomic molecules
and atomic lattices.
Consider the Taylor expansion of an
arbitrary potential function:
1
V ( x)  V0  V1  [ x  x0 ]  V2  [ x  x0 ]2  ...
2
Near a minimum, V1[xx0] ≈ 0.
Simple Harmonic
Oscillator
Consider the second-order term
of the Taylor expansion of a
potential function:
Letting x0 = 0.
V ( x)  12  ( x  x0 ) 2  12  x 2
Substituting this into
Schrödinger’s equation:
 2 d 2 ( x)

 V ( x) ( x)  E ( x)
2
2m dx

 m x 2 2mE 
d 2
2m   x 2
  2 
 E    2  2 
We have:
2
dx
 2



2mE
m
Let   2 and   2 , which yields:


2
d 2
2 2


x  
2
dx


The Parabolic
Potential Well
The wave function solutions
are:  n ( x)  H n ( x) exp( x 2 / 2)
where Hn(x) are Hermite
polynomials of order n.
n
|n |2
The Parabolic
Potential Well

Classically, the
probability of finding the
mass is greatest at the
ends of motion (because
its speed there is the
slowest) and smallest at
the center.


Classical
result

Contrary to the classical
one, the largest
probability for the lowest
energy states is for the
particle to be at (or near)
the center.
Correspondence Principle for the
Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase,
however, the solution approaches the classical result. This confirms the
Correspondence Principle for the quantum-mechanical simple
harmonic oscillator.
Classical
result
The Parabolic Potential Well
The energy levels are given by:
1
1
En  (n  )  / m  (n  )w
2
2
The zero point
energy is
called the
Heisenberg
limit:
1
E 0  w
2