THE SCHRODINGER WAVE EQUATION
UNBOUND STATES
The state of a system is described by the wavefunction  ( r , t ) . In the Schrodinger
representation, the wavefunction is a solution of the time dependent Schrodinger wave
equation
i

 r ,t  H  r ,t  ,
t
(1)
where H is a differential operator called the Hamiltonian. For a single particle of mass m
moving in a scalar potential energy field U ( r , t ) , the Hamiltonian is
2
 2
2
2 
H 
 2  2   U r ,t   
2  U  r , t  ,

2
2 m  x y z 
2m
and the normalization condition (probability of finding the particle is one) is
2
   r , t    r , t  dV  1 .
*
(2)
(3)
V
The probability density  associated with the single particle is
  r , t     r , t  ( r , t ) ,
*
(4)
and using equations (1) and (2), the time rate of change of the probability density gives
an equation of continuity
 *
 i
i

  *

* 2    2 *  
  *    * 

t
t
t 2 m
2m

  J  0,
(5)
t
where J is the probability current density
J
i
  *    *  .
2m
(6)
If   r , t  defines a pure energy or stationary state where the total energy of the
particle is E then
 r,t  r  e
a9/qm/doc/se equation.doc
i
Et
,
Saturday, January 03, 2009
(7)
1
and the time dependent Schrodinger equation (1) reduces to the time independent
Schrodinger equation
H   r   E  r  .
The wavefunction is not represented by the equation
 r,t  r  e
i
(8)
Et
,
because this equation implies that the total energy of the particle is negative.
RECTILINEAR MOTION OF A STREAM OF NON-INTERACTING PARTICLES
Free Particles
For a single particle moving along the x-axis as a free particle with total energy E and
zero potential energy, U(x) = 0, the time dependent Schrodinger equation is
2


  x, t   
  x, t  ,
t
2 m x 2
and has solutions of the form
(9)
i
  x, t    ( x ) e
i
Et
 Ae
i
2 mE
x
e
i
Et
 ( x )  Ae
i
2 mE
x
.
(10)

defines the total energy of the particle E (energy eigenstate) and
t
2

the spatial operator 
defines the momentum of the particle p  2 m E
2 m x 2
(momentum eigenstate). The total energy of the particle E determines the following
parameters describing the particle and its corresponding de Broglie wave
The time operator i
total energy E
kinetic energy K  12 m v 2  E
(E = K + U = K since U = 0)
2E
m
p  mv  k  2 m E
particle velocity v particle 
momentum
wave number (propagation constant) k 
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
2m E
2
de Broglie wavelength  
frequency
f 
period T 
h
E
2 h
h
 
k
p
2m E
E
h
angular frequency   2  f 
phase velocity v phase  f  

k
E

E
2m
d
2E

 v particle
dk
m
 2 v phase
group velocity vgroup 
vgroup  v phase
vgroup
probability current density (independent of time)
k *
J
 ( x )  ( x )  v particle  * ( x )  ( x ) .
m
The solution can be expressed as a monochromatic plane wave by
  x, t   Aei   k x  t  ,
(11)
where A is the amplitude of the wave. The + sign represents a wave traveling in the +x
direction and the – sign for traveling in the –x direction. The concept of a monochromatic
plane wave is an idealization and does not represent a real physical situation. The wave
extends from - to + and there is an infinite stream of particles each with energy E and
therefore the system has infinite energy.
It is assumed that the total energy of the particle E is exact. The momentum is
p  2 m E and hence the momentum is precisely defined. Therefore the particle is not
localized and its position is completely uncertain. The wavefunction   x, t  can not be
normalized and  * ( x )  ( x ) can not represent the probability density. However, it can be
interpreted as the particle density of a stream of non-interacting particles each moving
with total energy E   and momentum p  k .
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
3
In the S.I. system of units: the wavefunction (x) has units of m-0.5;  * ( x )  ( x ) has
units of m-1; the velocity vparticle has units of m.s-1; and the current probability density J
has units of s-1. Therefore the current probability J can be interpreted as the number of
particles n passing a point along the x-axis per second
J n
k *
 ( x )  ( x )  v particle  * ( x )  ( x ) .
m
(12)
For a uniform particle beam, the amplitude A is related to the intensity of the beam. The
number of particles NL on average in a length L is
xL
xL
x
x
NL  
 * ( x) ( x) dx  
A2 dx A2 L
Therefore, A2 can be interpreted as the number of particles on average per meter.
The time independent Schrodinger equation in the case of rectilinear motion in a potential
energy field U(x) is
 2  x 

 U ( x )  E   x   0 .
2 m x 2
2
If the potential energy U(x) has a finite discontinuity at x0 so will
(13)
 2  x 
but   x  and
x 2
  x 
need to be continuous to give physically acceptable solutions.
x
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
4
MATLAB – MOTION OF FREE PARTICLE
plane_wave_A.m
The m-script plane_wave_A.m animates the solution of the time dependent Schrodinger
equation for a stream non-interacting particles (electrons) moving along the x-axis with
total energy E per particle. Examine and run the m-script for the default values (E = 100
eV). Make sure you understand what the m-script tells you and how the calculations are
performed.
To understand the motion of the particle it is necessary to calculate and plot both the real
and imaginary parts of the wavefunction. It is also useful to display how the phase of the
wavefunction evolves with time. The time independent wavefunction   x  is only
calculated once and the time development of the wavefunction is then given by the
multiplication of   x  by e
form
i
Et
 e  i  t as described by equation (11) expressed in the
  x, t   Ae i k x ei  t .
For particles with energy E = 100 eV, the Matlab graphical output and numerical output
are:
Wavefunction: Stream of non-interacting particles
100
psi ( m -0.5 )
Investigations
real
imag

0
phase (  rad )
-100
-2
-1
0
x (m)
1
2
x 10
-10
2

1
wavelength: phase cycles through 2
0
-2
x 10
-1
0
x (m)
1
2
x 10
-10
10
-1
J (s )
10
5
constant probability current density for free particles
0
-2
-1
0
x
(
m)
--------------------------------------------------------------------------------------------
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
1
2
x 10
-10
5
Paramaters for stream of non-interacting particles and associated de Broglie matter wave
-------------------------------------------------------------------------------------------Energy of each particle, E = 1.000e+002 eV
Kinetic energy of each particle, K = 1.000e+002 eV
Momentum of each particle, p = 5.402e-024 N.s
Velocity of each particle (group velocity of de Broglie wave), v_gp = 5.931e+006 m/s
Wave number of de Broglie wave, k = 5.12e+010 rad/m
de Broglie wavelength, lambda = 1.227e-010 m
Angular frequency of de Broglie wave, omega = 1.518e+017 rad/s
Frequency of de Broglie wave, f = 2.417e+016 Hz
Period of matter wave, T = 4.138e-017 s
Phase velocity of de Broglie wave, v_ph = 2.964e+006 m/s
Amplitude of de Broglie wave, A = 1.000e+002 m^(-0.5)
Average number of particles per meter along x-axis, N = 1.000e+004
Number of particles passing a point along x-axis per second, n = 5.931e+010 /s
Exercises
1.
2.
3.
4.
Run the program a few times and change the animation time. Comment on each
graph.
Calculate “manually” each of the parameters in numerical output to verify that
they are correct.
Predict the changes in the parameters when the total energy of each electron is
increased from 100 eV to 400 eV. Check your predictions.
Use the data cursor and the ginput Matlab command to estimate the wavelength 
from the (x) graph and the phase angle graph. Do the estimates agree with the
theoretical value?
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
6
Stream of particles incident at a potential energy step with E > U0
Consider the motion of a stream of non-interacting particles in a potential energy field
U(x) where in region 1, U(x) = 0 for x < 0 and in region 2, U(x) = U0 (constant) > 0 and
U0 < E.
energy
E
I
U = U0
R
T
x=0
U=0
+x
Fig. xx. The total energy E and potential energy U(x) function for an electron
incident upon a potential step with step height less than the total energy.
The time independent Schrodinger equation (12) is
 2  x 
 k 2   x 
2
x
region 1
U(x) = 0
x<0
k 2  k12 
region 2
U(x) = U0
x>0
k 2  k2 2 
2m E
2
2 m (E  U0 )
2
This corresponds to the classical case where each particle has sufficient energy to
surmount the step potential energy barrier and pass into region 2. The step barrier is a
repulsive for electrons, hence the kinetic energy of the electrons crossing the step is
reduced (K = E – U0). Therefore, the momentum of each electron is smaller and the de
Broglie wavelength is longer.
We can consider the time dependent wave function to be
region 1
x<0
region 2
x>0
1  x, t    AI ei k1 x  AR e  i k1 x  e  i  t
 2  x, t    AT ei k2 x  e  i  t
where 1  x, t  represents the superposition of the incident wave (amplitude AI) and the
reflected wave (amplitude AR) and  2  x, t  defines the transmitted wave.
The boundary conditions at x = 0 are
1  0, t    2  0, t 
1  0, t   2  0, t 

x
x
and these conditions give
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
7
k1  AI  AR   k2 AT
AI  AR  AT
and hence
AR 
k1  k2
AI
k1  k2
AT 
2 k1
AI
k1  k2
or
(14)
AI 
k1  k2
AT
2k1
AR 
k1  k2
AT
2k1
Normally one specifies the amplitude of the incident wave. However, in the following
Matlab simulations, it is more desirable to specify the amplitude of the transmitted wave
AT. The number of particles incident nI, reflected nR and transmitted nT per second at the
barrier (x = 0) are determined from equation (12) where nI = nR + nT
k
nT  2 AT 2
m
2
2
k k k 
nI  1  1 2  AT 2
m  2k1 
k k k 
nR  1  1 2  AT 2
m  2k1 
2
2
(15)
k k 
k k 
nI  v1  1 2  AT 2
nR  v1  1 2  AT 2
 2k1 
 2k1 
where v1 and v2 are the speeds of the particles in regions 1 and 2 respectively The
coefficient of reflection R and coefficient of transmission T (R+T = 1) are thus given by
nT  v2 AT 2
2
k k  v v 
n
R R  1 2  1 2
nI  k1  k2   v1  v2 
2
T
nT
4 k1 k2
4 v1 v2


2
2
nI  k1  k2 
 v1  v2 
(16)
Stream of particles incident at a potential energy step with E < U0
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
8
k ( rad/s )
6
E = 100 eV > U
0
4
k2
2
4
 (m)
Potential Step Barrier
10
k1
0
0
x 10
20
40
60
80
6
100
-9
1
2
2
0
v ( m/s )
x 10
0
x 10
20
40
60
80
100
6
v1
4
v2
2
0
0
Fig. xx.
20
<--- large step
40
60
E - 0U( eV )
80
small step -->
100
Variations in k,  and v with the height of the step.
step_potential.m
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
9
Potential Step Barrier E = 100 eV > U0
100
A ( m-0.5 )
AI
AR
50
0
particles n / s
6
0
x 10
20
40
60
100
10
nI
4
nR
2
nT
0
0
20
40
60
1
R and T
80
80
100
R
T
R+T
0.5
0
0
Fig. xx.
20
<--- large step
40
60
E - 0U( eV )
80
small step -->
100
Variations in A, n, R and T with the height of the step potential.
step_potential.m
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
10
a9/qm/doc/se equation.doc
Saturday, January 03, 2009
11