I
incident
reflected


i
  x, t   
  x, t   V  x , t    x , t 
2
t
2m x
2
2
II
transmitted
En   n  12  
Heisenberg’s Matrix Mechanics
1924: de Broglie suggests particles are waves
Mid-1925: Werner Heisenberg introduces Matrix Mechanics
•Semi-philosophical, it only considers observable quantities
•It used matrices, which were not that familiar at the time
•It refused to discuss what happens between measurements
•In 1927 he derives uncertainty principles
Late 1925: Erwin Schrödinger proposes wave mechanics
•Used waves, more familiar to scientists at the time
•Initially, Heisenberg’s and Schrödinger’s formulations were competing
•Eventually, Schrödinger showed they were equivalent; different descriptions which
produced the same predictions
Both formulations are used today, but Schrödinger is easier to understand
The Free Schrödinger Equation
1925: Erwin Schrödinger proposes wave mechanics
•Peter Debye suggested to him he needed to find a wave equation
for quantum mechanics
2
p
•He hit on the idea of using complex waves
E
•The rest is history
2m
  x, t 
2
•Starting point: Energy/Momentum relationship
p
•Multiply by the wave function on the right
E   x, t  
  x, t 
2m 2
•Use de Broglie relations to rewrite
E 
p k
  x, t  
•Use relationships for complex waves to rewrite with derivatives

 ik
x

 i
t
1 
k
i x

 i
t
k   x, t 
2
2m
  x, t  exp  ikx  it 


i
  x, t   
  x, t 
2
t
2m x
2
2
Sample Problem with Free Schrödinger
Show that the following expression satisfies the free
Schrödinger equation, and find the constant A:
2

2
i
  x, t   
  x, t 
2
t
2m x
2

1
Aix 
  x, t  
exp 

t
 t 
2
2
2







 1
Aix
 1 
Aix
1 
Aix 
exp 
 exp 
     exp 


t  t
t t
 t   t  t 
 t 
 t 
 Aix 2  1
 Aix 2  iAx 2 
1
  3/2 exp 
  exp 
 2 
2t
t
t
t



 t 
Sample Problem with Free Schrödinger (2)
Show that the following expression satisfies the free
Schrödinger equation, and find the constant A:
2

2
i
  x, t   
  x, t 
2
t
2m x
2

1
Aix 
  x, t  
exp 

t
 t 
m  it  2 Ax 2   2 A  it  2 Ax 2 
2
2





 1
Aix
2 Aix
Aix 
exp 
 exp 
 

x  t
 t  t t
 t 
m
A
2
 Aix 2    2 Ai 4 A2i 2 x 2 
 Aix 2 
2  1
exp 
 2
exp 
  


2 
x  t
t t 
 t   t t
 t 
2
 1
 2 Ai 4 A2i 2 x 2 
iAx 2 
i   3/2  2   
 2


2
t
2
m
t t
t t 

t t
Multiply by 2mt5/2/
The Schrödinger Equation
The General Prescription for Classical  Quantum:
1. Write a formula for the energy in terms of momentum and position
2. Transform Energy and momentum using the following prescription:

E   i

t
3. Rewrite it as a wave equation

p k
i x
What if we have forces?
•Need to add potential energy V(x,t) on top of kinetic energy term
2
2

2
i

 V  x, t 
2
t
2m x
p
E
V  x, t 
2m
2

2
i
  x, t   
  x, t   V  x , t    x , t 
2
t
2m x
2

 2
i

V 
2
t
2m x
Comments on Schrödinger Equation
2
1. This equation is inherently

2
i
  x, t   
  x, t   V  x , t    x , t 
complex
2
t
2m x
•You MUST use complex
Initial conditions:
wave functions
Classical physics
2. This equation is first order in time
x(t = 0) and v(t = 0)
•It has only first derivatives with respect to time
•If you know the value at t = 0, you can work it out at subsequent times
•Proved using Taylor expansion:
  x, t       x, t       x, t   12  2    x, t  
2
t
t

   x, t      x, t 
Initial conditions:
t
Quantum physics
 2 2


(x,t = 0)
   x, t   
  x, t   V  x, t    x, t 
2
i  2m x

2
The Superposition Principle
2

2
i
  x, t   
  x, t   V  x , t    x , t 
2
t
2m x
3. This equation is linear
•The wave function
appears to the first power everywhere
•You can take linear combinations of solutions:
  x, t 
1  x, t 
 2  x, t 
i
 c1i
 c2i
t
t
t
Let 1 and 2 be two solutions
of Schrödinger. Then so is
  x, t   c11  x, t   c22  x, t 
where c1 and c2 are arbitrary
complex numbers
2
2




 2 1  x , t 
 2  2  x, t 
 c1  
 V  x, t  1  x, t    c2  
 V  x, t   2  x, t  
2
2
x
x
 2m

 2m

2
2

c  x, t   c2  2  x, t    V  x, t  c11  x, t   c2  2  x, t  
2  1 1
2m x
 2   x, t 

 V  x, t    x , t 
2
2m
x
2
Q.E.D
Time Independent problems
2
2


•Often [usually] the
i
  x, t   
  x, t   V  x    x , t 
2
potential does not
t
2m x
depend on time: V = V(x).
 x, t     t   x 
•To solve this equation, we try separation of variable: 
2
2
•Plug this guess in:


i   x   t   
  t  2   x  V  x   t   x 
•Divide by the
t
2m
x
original wave function
•Note that left side is independent of x, and right side is independent of t.
•Both sides must be independent of both x and t
•Both sides must be equal to a constant, called E (the energy)
2
d  t 
d 2  x 
 E 
V  x
2
  t  dt
2m  x  dx
i
Solving the time equation
  x, t     t   x 
d  t 
d   x
E
V  x
2
  t  dt
2m  x  dx
2
2
i
•We have turned one equation into two
•But the two equations are now ordinary differential equations
•Furthermore, the first equation is easy to solve:
d  t  Edt

 t 
i

d


iE
ln   t   
 dt
•These types of solutions are called stationary states
•Why? Don’t they have time in them?
•The probability density is independent of time
  x, t   e
2
 iEt
  x e
iEt
iEt
  t   eiEt
  x, t   eiEt   x 
 * x    x
2
The Time Independent Schrödinger Eqn
d 2  x 
E
V  x
2
2m  x  dx
2
d 2  x 
E  x   
 V  x   x 
2
2m dx
2
Multiply by (x) again
2
2
d

•This equation is much easier to solve than the
E  
 V
2
2m dx
original equation
•ODE’s are easier than PDE’s
•It can pretty easily be solved numerically, if necessary
•Note that it is a real equation – you don’t need complex numbers
•Imagine finding all possible solutions n(x) with energy En
•Then we can find solutions of the original Schrödinger Equation
  x, t    n  x  exp   iEnt

•The most general solution is superposition of this solution
  x, t    cn n  x  exp   iEnt
n
