```Free Space
E is a fixed quantity greater than 0. The Schrödinger equation is solved by minimizing
  2  x, c , E 

 E  x, c   dx
10  
2
 (1.1)
 2 c, E  
2
10
10
   x, c , E  dx
2
10
Try the form
  x, k , E   exp  ikx 
k2
E 0
2
k   2E
(1.2)
So that a pulse of particles is given by
  x, t  
 

k2
2
exp

k

k
exp
i
k
x

x





 t  t0   dk (1.3)


0
0

2

 



Figure 1 k0 = 7, x0 = -7, t0 = -1. the three files are the wave function at t = -1, 0, 1
The integration method is described in ..\..\definitions\AiGau\GuidingFunction.docx. Twenty points were
generated for a midpoint trap rule and the integrand was summed over these. The code for this is in
TWave1.zip.
The wave function starts narrow and expands as time passes. All units are in Hartrees.
Figure 2 The energy is averaged over exp(-((E-E0)/w)2) E0 = 10, w=0.7 The times ar 0, 8, 16
Figure 2 and 3 are based on Twave2.zip. This file selects the energy from a Gaussian with variable widths.
The methodology is described in ..\..\definitions\AiGau\GuidingFunction.docx.
Figure 3 w changed from 0.7 to 1.4.
One Barrier
Figure 4 Box barrier
For E > V, there are two regions
1. To the left f -1 and to the right of +1
  x, k1 , E   exp  ikx 
k12
E 0
2
k1   2 E
2.
(2.1)
In the center
  x, k2 , E   exp  ikx 
k2 2
V  E  0
2
(2.2)
k2   2  E  V 
This is 6 complex wave function forms with each complex coefficient in the form cnexp(icn+1)
1.
2.
3.
c1 exp  i  c2  k1 x  Et  
c3 exp  i  c4  k1 x  Et  
(2.3) to the left of -1
c7 exp  i  c8  k2 x  Et  
(2.4) from -1 to +1
c5 exp  i  c6  k2 x  Et  
c9 exp  i  c10  k1 x  Et  
c11 exp  i  c12  k1 x  Et  
(2.5) to the right of +1
The boundary conditions are that the derivatives match at ± 1. In Bohm’s example [11.5 pp 23-240], the
incoming wave is from the left so that c11 is zero. Arbitrary normalization allows us to assume that c1 is
equal to 1 and c2 equal to zero. The value of k2 is imaginary for V > E. We will stay epsilon away from V
= E.
  2  x, c , E 



V

E

x
,
c



 dx

2
10 

2
 c, E  
(2.6)
2
10
10
   x, c , E  dx
2
10
The variance minimization above seems useless. The Schroedinger equation for the above wave function is
exactly satisfied in each region. Interesting things happen at ±1.
The numerical second derivative of ∅ at ±1 is
 ''  t0 ,   
 '  t0      '  t0   
(2.7)
2
So that in this case the variance redueces to
 1 d 2  x, c , E 

 V  E  x, c   dx

2

2
dx

 2  c , E   1 
2
1
1
    x, c , E  dx
2
1
1
 1 d   x, c , E 

 V  E  x, c   dx
2
dx

   2
 1
2
2
(2.8)
1
   x, c , E  dx
2
1
For small enough delta, this reduces to the usual boundary condition that at exactly ±1, ∅’ and ∅ must be
continuous.
In the case that E is less than V, the k2 changes
Equation (2.4) can be thought to be a slow motion travelling wave, equation
Error! Reference source not found. just decays.
Two Barriers
Figure 5 Two Barriers with a space
For E > V, there are three regions
3. To the left of -3 and to the right of +3 and between -1 and 1
  x, k1 , E   exp  ikx 
k12
E 0
2
k1   2 E
4.
(2.9)
Between -3 and -1 and between 1 and 3.
  x, k2 , E   exp  ikx 
k2 2
V  E  0
2
(2.10)
k2   2  E  V 
Variance Minimization
In 1d, V(x)= 1-cos(x),  = /x,
Figure 6 The potential is not that of the usual harmonic oscillator but it expands as x2 for small |x|.
  2  x , c , E 



V
x

x
,
c
,
E

E

x
,
c







 dx

2
10 
2

 c, E  
2
10
(3.1)
10
   x, c , E  dx
2
10
A trial wave function is
exp  i  pn x   1  c1 AiGau  c2 x   


  x, c    exp  i  pn x    AiGau (c 2 x  


2
c3 exp  c4 x 

(3.2)
The first term is an incoming plane wave modulated by the potential. The second term is the reflected
plane wave. The third term is the trapped harmonic oscillator.
The qualitative difference between this system and one with a negative potential is that there is a
solution for all positive energy values. This allows packets of nearly equal wave functions to account for
the particle aspects of the system.
Barrier Penetration
Figure 7 The rectasngular well approximates the potential. (the potential is in Hartrees)
This system is treated in Bohm 11.5. The energy E is less than 1.
To the right
R  x, E   A exp  ip1 x  x  1
p1  2 E
(4.1)
Inside the barrier
I  x, E   B exp  p2 x   C exp   p2 x   1  x  1
p2  2 V  E 
(4.2)
To the left
R  x, E   D exp  ip1 x   F exp  ip1 x  x  1
p1  2 E
(4.3)
The full wave function is found by matching the values of the wave function and its derivatives at ± 1.
The derivatives introduce imaginary terms making the coefficients complex.
The resulting transmission coefficient is
T
A
D
2
2
Attractive Square Well
Figure 8 The potential has been moved down a unit.
This system is treated in Bohm 11.71
To the right
R  x, E   A exp  ip1 x  x  1
p1  2 E
(5.1)
Inside the barrier
I  x, E   B exp  ip2 x   C exp  ip2 x   1  x  1
p2  2  E  V 
(5.2)
To the left
R  x, E   D exp  ip1 x   F exp  ip1 x  x  1
p1  2 E
(5.3)
This is again solved by matching boundary conditions leading to
1
David Bohm, Quantum Mechanics, Prentice Hall (1951) pp 242-243
T
1
1
2
1   p1 / p2  p2 / p1  sin 2  2 p2 
4
(5.4)
Figure 9 Transmission versus incoming energy.2
The code for this is in ResTrans.zip
Time delay
Bohm introduces time dependence into the wave function by constructing a wave packet.
 packet  x, t    dp1 f  p1  p0    p1  exp  iE  p1  t  (6.1)
The integral over p1 averages the wave function to zero except for those value of p+1+ with relatively the
same phase for each term.
I the region of the incident wave
  p1   exp  ip1 x 
(6.2)
The phase is
Phase  i  p1 x  Et  (6.3)
The maximum value of x occurs when this has zero derivative with respect to p 1
2
This barely resembles the plot in Bohm.

 p1 x  E  p1  t  p1  p0  0
p1
(6.4)
2
E  p1   p1 / 2
Or
x  p0 t
The complex transmitted part of ∅ can be written as
A exp   so that the transmitted wave function
becomes
 packet  x, t    dp1 A f  p1  p0  exp  i  p1 x    E  p1  t   (6.5)
This has a maximum at
 E 
  
x
t 
(6.6)


 p1  p1  p0
 p1  p1  p0
This second term is a tme delay caused by the potential. It comes from the phase of the complex
coefficient A.
Meta Stable state
Bohm [11.14 p 284] introduces a meta stable state that he “solves” in the WKB approximation.
This state superficially resembles figure 1.
The randomness needed to average the frequencies is given this time by
 packet  x, t    dp1 f  E  EN    E  exp  iEt dE
(7.1)
The switch from p to E is a inconsistent, but serves to emphasize the fact that all that the f does is average
over slightly different wave functions. If E were not > 0, in most potential forms, there would be no nearby
∅’s to average over. The wave function that matches the plane waves on its two edges is complex
with a real plus imaginary part. A lot of algebra and a few approximations yields a value for ∆t and
2
J  2  p1 dx
(7.2)
J
E
(7.3)
0
0 
  exp
  p dx 
t   0  2
2
0
(7.4)
1
(7.5)

exp(iEt )
dE
1  i  E  EN  t

 packet  x, t     En  
exp  t / t t  0
   En  
t0
0
(7.6)
Harmonic Oscillator
Try
  x, c1   exp  c1 x 2 
  x, c1 
(7.7)
 2c1 x exp  c1 x 2 
x
 2  x, c1 
  2c1  4c12 x 2  exp  c1 x 2 
2
x
So that (3.1) becomes
  c
1
1
 2 c, E  

 2c12 x 2   1  cos  x    E  c  exp  2c1 x 2 dx
1
2
(7.8)
1
 exp  2c x  dx
2
1
1
Expanding the cosine as 1-x2/2, the integrand becomes independent of x for
2c12 x 2 
x2
0
2
or
c1 
(7.9)
1
2
Then if E(c1)=c1, the integrand, and σ2, become 0.
Resonance
The periodic, complete, orthonormal function in a distance L is
 2 nx 
exp  i
 (8.1)
L
 L 
1
Following Preston at x0 equate the logarithmic derivatives
2c1 x0  i
2 n
L
An arbitrary function can be expanded as
  x, c , t  
1
L
 n 2 x 
 a  c , t  exp  i L  (8.2)

n 
n
The time development is given by
i
  x, c, t 
Or
t
 2

 
 V  x     x, c, t  (8.3)
 2

  n2 2

an
 n2 x 
 n2 x 
n i t exp  i L   n  2L2  V  x   an exp  i L  (8.4)


Multiply by
 n ' 2 x 
exp  i
 and integrate using the orthogonality of the exponentials
L 
L

1
 n2 
a
i n 
  an 'V  n  n ' (8.5)
t
2 L2
n'
2
Where
  n  n ' 2 x 
V  n  n '   exp  i
 dx (8.6)
L


```