Lecture 10-11
Schrödinger (薛定谔)equations
Prior to 1925 quantum physics was a “hodgepodge” of
hypotheses, principles, theorems and recipes.
It was not a
logically consistent theory.
Once we know this wavefunction we know “everything”
about the system!
Outline
Part 1 Dynamic Equations
Part 2 Dynamic Equation of Wave functionTime-dependent Schrödinger equation (TDSE)
Part 3 Stationary state Schrödinger equation (TISE)
Part 4 Conditions on wave function and Probability
current density
Part 1 Dynamic Equations
If we know the forces acting upon the particle than, according to
classical physics, we know everything about a particle at any
moment in the future.
r
r
r
r
r
r
r
r
d 2r
F  ma , F  U (r ),  U (r )  m 2
dt
r r
2
r
r
1

E
(r )
 2 E (r )  2
0
c
t 2
A differential equation by itself does not fully determine the
r r
r
unknown function r (t ) or E (r , t ) .
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 1 of 18
Part 2 Dynamic Equation of Wave function
---- Schrödinger equations
用  ( x) 
1  2
1

sin
kx

cos kx  描述的粒子，只能有5种动量取值，分别是

2

h 
1 1 1 1 1
2 8 8 8 8
0, 2k ,  2k , k ,  k ，对应的几率分别是 , , , , ，这些几率总和应该为1。
k
1
1
1
1
1
p   P( pi ) pi  0   2kh    2kh    kh     kh    0,
2
8
8
8
8
i 1
k
1 1 1 1 1
P
(
p
)

    1

i
2
8 8 8 8
i 1
k
k
i 1
i 1
x   P( xi ) xi ,  P( xi )  1  x   p( x) xdx,  p( x) dx  1
x    ( x) xdx,
2
¡
¡

¡
¡
 ( x) dx  1
2
x    ( x) xdx    * ( x) ( x) xdx    * ( x) x ( x)dx
2
¡
¡
¡
Do we have the same recipe for calculation of average
momentum
by
using
wave
function
in
position
representation? Yes, of course, we have!
To find the expectation (average) value of p, we first need
to represent p in terms of x and t. Consider the derivative
of the wave function of a free particle with respect to x:
 p0 ( x, t ) 
1
i

exp  ( p0 x  E0t ) 
2 h
h

We find that
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 2 of 18

i

 p0 ( x, t )  p0  p0 ( x, t )  ih  p0 ( x, t )  p0  p0 ( x, t )
x
h
x

p  p0    *p0 ( x, t )(ih ) p0 ( x, t )dx  p0   *p0 ( x, t ) p0 ( x, t )dx  p0 (0)
¡
¡
x
This suggests we define the momentum operator as

p̂  ih
x
The expectation value of the momentum is
p    * ( x) pˆ ( x)dx    * ( x)(ih
¡
¡

) ( x)dx
x
1  2
1

sin kx  cos kx 

2

h 
 C00 ( x)  C2 kh  2 kh ( x)  C2 kh  2 kh ( x)  Ckh  kh ( x)  C kh   kh ( x)
 ( x) 
=
1
1
1
1
1
0 ( x) 
 2 kh ( x) 
 2 kh ( x) 
 kh ( x) 
  kh ( x)
2
8
8
8
8
px
i
1
 p ( x) 
eh
2 h
k
1
1
1
1
1
p   P( pi ) pi  0   2kh    2kh    kh    kh    0
2
8
8
8
8
i 1
So, we can not have definite values for the dynamical
variables, such as the momentum, when the state of a
particle is determined by the wave function with respect to
x.
We have to find the other way to describe the
dynamical variables in Quantum Mechanics.
For every dynamical variable or any observable there
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 3 of 18
is a corresponding Quantum Mechanical Operator
Physical Quantities  Operators
Operators are important in quantum mechanics.
All observables have corresponding operators.
Operators  Symbols for mathematical operation
 The position x is its own operator x̂  x . Done. Other operators

2
are simpler and just involve multiplication x  x 2  x  x .
 The potential energy operator is just multiplication by V(x).
 The momentum operator is defined as p̂  ih
Momentum px  pˆ x  ih

x

x
1
Kinetic Energy T  Tˆ 
pˆ x pˆ x
2

Total Energy E  Eˆ  ih
t
Position x  xˆ  x
Potential Energy U ( x)  Uˆ ( x)  U ( x)
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 4 of 18
 p ( x) 
0
i
1
e
2 h
p0 x
h
p x
p x
i 0
i 0 
ip
  1

i
h
0
pˆ x p0 ( x)  ih 
e h  p0
e h 
x  2 h
2 h h

pˆ x p0 ( x)  p0 p0 ( x)
i
1
e
2 h
p0 x
h
p0 2
1
1
ˆ
T  p0 ( x) 
pˆ x [ pˆ x p0 ( x)] 
pˆ x ( p0 p0 ( x)) 
 p0 ( x)
2
2
2
p0 2
ˆ
T  p0 ( x) 
 p ( x)  T0 p0 ( x)
2 0
p x
i 0
1
xˆ p0 ( x)  x
e h  x p0 ( x)
2 h
1
i

 p0 ( x, t ) 
exp  ( p0 x  E0t ) 
2 h
h

 1
i

 iE 
Eˆ  p0 ( x, t )  ih 
exp  ( p0 x  E0t )    ih   0   p0 ( x, t )
t  2 h
 h 
h

Eˆ  ( x, t )  E  ( x, t )
p0
0
p0
Eigenvalue equation of an operator
aˆn  nn
operator
Created by Tianxin Yang, College
eigenfunction
eigenvalue
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 5 of 18
Deriving the Schrödinger Equation using operators:
This was a plausibility argument, not a derivation.
We
believe the Schrödinger equation not because of this
argument, but because its predictions agree with
experiments.
p2
 U ( x)  E
2m
Created by Tianxin Yang, College
h 2    x, t 



U
(
x
)

x
,
t

i
h
  x, t 


2m  2 x
t
2
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 6 of 18
Schrödinger Equation Notes:
The Schrödinger Equation is THE fundamental equation of
Quantum Mechanics. There are limits to its validity.
In this form
it applies only to a single, non-relativistic particle (i.e. one with
non-zero rest mass and speed much less than c.)
系综平均（Ensemble Average）

On the left hand picture 13 velocity vectors of an individual fly are shown; the chain
of vectors closes so vit = 0.

On the right hand picture the same 13velocity vectors are assigned to 1 fly each to
demonstrate that the ensemble average yields the same result, i.e. ve = 0,
provided that each and every fly does the same thing on average.

i.e. time average = ensemble average. The new subscripts "e" and "r" denote
ensemble and space, respectively. This is a simple version of a very far reaching
concept in stochastic physics known under the catch word "ergodic hypothesis".
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 7 of 18

As long as every fly does - on average - the same thing, the vector average over
time of the ensemble is identical to that of an individual fly - if we sum up a few
thousand vectors for one fly, or a few million for lots of flies does not make any
difference. However, we also may obtain this average in a different way:

We do not average one fly in time obtaining vit , but at any given time all flies in
space.

This means, we just add up the velocity vectors of all flies at some moment in time
and obtain ver , the ensemble average. It is evident (but not easy to prove for
general cases) that
vit = ve
a) Schrödinger equation is a linear homogeneous partial
differential equation.
2
h 2    x, t 



U
(
x
)

x
,
t

i
h
  x, t 


2
2m  x
t
b) The Schrödinger equation contains the complex number i.
Therefore its solutions are essentially complex (unlike
classical waves, where the use of complex numbers is
just a mathematical convenience.)
c) The wave equation has infinite number of solutions, some
of which do not correspond to any physical or chemical
reality.
1. For an electron bound to an atom/molecule, the wave
function must be everywhere finite, and it must vanish in the
boundaries
2. Single valued
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 8 of 18
3. Continuous
4. Gradient (d/dr) must be continuous
5.  *d is finite, so that  can be normalized
d) Solutions that do not satisfy these properties (above) DO
NOT generally correspond to physically realizable
circumstances.
e) Conditions on the wave function(波函数的三个基本条件—
—有限、单值、连续)
1. In order to avoid infinite probabilities, the wave function
must be finite everywhere.
2. The wave function must be single valued.
3. The wave function must be twice differentiable.
This
means that it and its derivative must be continuous.
(An
exception to this rule occurs when V is infinite.)
4. In order to normalize a wave function, it must approach
zero as x approaches infinity.
f)
Only the physically measurable quantities must be real.
These include the probability, momentum and energy.
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 9 of 18
h 2    x, t 



U
(
x
)

x
,
t

i
h
  x, t 


2m  2 x
t
2
Can think of the LHS of the Schrödinger equation as a differential
operator that represents the energy of the particle? This operator
is called the Hamiltonian of the particle, and usually given the
symbol
Ĥ . Hamiltonian is a linear differential operator.
 h2 d2

ˆ


V
(
x
,
t
)
 2m dx 2
   H


Created by Tianxin Yang, College
Kinetic
Potential
energy
energy
operator
operator
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 10 of 18
Hence there is an alternative (shorthand) form for the
time-dependent Schrödinger equation:
ih

  x, t   Hˆ  x, t 
t
Part 3 Time-independent Schrödinger equation (TISE),
i.e. stationary state(定态) Schrödinger equation
Suppose potential is independent of time
U  x, t   U ( x )
Look for a separated solution, substitute
 ( x, t )   ( x)T (t ) into
h 2    x, t 



U
(
x
)

x
,
t

i
h
  x, t 


2m  2 x
t
2
h 2 2



(
x
)
T
(
t
)

U
(
x
)

(
x
)
T
(
t
)

i
h

 ( x)T (t )
2 
2m x
t
h 2 1 d 2 ( x)
1 dT (t )


U
(
x
)

i
h
E
2
2m  ( x) dx
T (t ) dt
ih
•
1 dT (t )
 E  T (t )  aeiEt / h
T (t ) dt
This only tells us that T(t) depends on the energy E. It
doesn’t tell us what the energy actually is. For that we have to
solve the space part.
•
T(t) does not depend explicitly on the potential U(x). But
there is an implicit dependence because the potential affects
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 11 of 18
the possible values for the energy E.
h 2 1 d 2 ( x)

 U ( x)  E
2m  ( x) dx 2
h 2 d 2 ( x)

 U ( x) ( x)  E ( x)  Hˆ  ( x)  E ( x)
2
2m dx
2
2
h
d
Hˆ  
 U ( x)
2m dx 2
 h2 d2

ˆ
H ( x)  E ( x)   

U
(
x
)
 ( x)  E ( x)
2
2
m
dx


This is the time-independent Schrödinger equation (TISE) or
so-called stationary state Schrödinger equation.
Solution to full TDSE is
( x, t )   ( x)T (t )   ( x)eiEt / h
Even though the potential is independent of time the wavefunction
still oscillates in time.
But probability distribution is static
P  x, t     x, t    * ( x)e  iEt / h ( x)e  iEt / h
2
  * ( x) ( x)   ( x)
2
For this reason a solution of the TISE is known as a
Stationary State(定态)
Stationary state Schrödinger Equation Notes:
• In one-dimension space, the TISE is an ordinary differential
equation (not a partial differential equation)
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 12 of 18
• The TISE is an eigenvalue equation for the Hamiltonian
operator: Hˆ  ( x )  E ( x )
Part 4 Probability current density and continuity equation
Definition of probability current density
In non-relativistic quantum mechanics, the probability current
of
the wave function Ψ is defined as
in the position basis and satisfies the quantum mechanical
continuity equation
with the probability density
defined as
.
If one were to integrate both sides of the continuity equation with
respect to volume, so that
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 13 of 18
then the divergence theorem implies the continuity equation is
equivalent to the integral equation
where the V is any volume and S is the boundary of V. This is the
conservation law for probability in quantum mechanics.
In particular, if
is a wavefunction describing a single particle,
the integral in the first term of the preceding equation (without the
time derivative) is the probability of obtaining a value within V
when the position of the particle is measured. The second term is
then the rate at which probability is flowing out of the volume V.
Altogether the equation states that the time derivative of the
change of the probability of the particle being measured in V
is equal to the rate at which probability flows into V.
Derivation of continuity equation
The continuity equation is derived from the definition of probability
current and the basic principles of quantum mechanics.
Suppose
is the wavefunction for a single particle in the
position basis (i.e.
Created by Tianxin Yang, College
is a function of x, y, and z). Then
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 14 of 18
is the probability that a measurement of the particle's position will
yield a value within V. The time derivative of this is
where the last equality follows from the product rule and the fact
that the shape of V is presumed to be independent of time (i.e. the
time derivative can be moved through the integral). In order to
simplify this further, consider the time dependent Schrödinger
equation
and use it to solve for the time derivative of
:
When substituted back into the preceding equation for
this
gives
.
Now from the product rule for the divergence operator
and since the first and third terms cancel:
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 15 of 18
If we now recall the expression for P and note that the argument
of the divergence operator is just
this becomes
which is the integral form of the continuity equation.
The differential form follows from the fact that the preceding
equation holds for all V, and as the integrand is a continuous
function of space, it must vanish everywhere:
For all whole space we have
 2 
r r
lim 
 dV   lim    j dV  0
V   
V 
t 
V 
V
r r
r 
lim    j dV  lim  j  ds  0

V 

V
which means that

S 

S
must be continuous at any position in
the whole space.
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 16 of 18
So the wavefunction and its derivative must be continuous.
(An exception to this rule occurs when V is infinite.)
current
means
 ( x)
i
Et
( x, t )   ( x)e h and  ( x ) is real, the probability
r
r h
j  Im  * ( x) ( x)   0 over the whole 1D space which

m 
One more, if the
r
j
is always continuous whatever the wavefunction
and its derivative  ( x) are continuous or not.
 ( x ) has
However,
to be continuous for an acceptable physical solution
for that the probability density is uniquely defined(唯一确定).
As to  ( x) , it may not be continuous especially at the point
where the potential energy is infinite.
It is easy to prove that  ( x) has to be continuous at the
point x0 where the potential energy just has a limited high
step.
2
2


h
d
Hˆ  ( x)  E ( x)   

U
(
x
)
 ( x)  E ( x)

2
 2m dx

Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 17 of 18
h 2 d 2 ( x)

 U ( x) ( x)  E ( x)
2
2m dx
x0 
h 2 x0 



(
x
)
dx


U ( x)  E  ( x)dx



x


x


0
2m 0
If U ( x)  E  is not infinite at the point of x0 ,
lim 
x0 
lim 
x0 
 0 x0 
 0 x0 
U ( x)  E  ( x)dx  0
 ( x)dx  lim  ( x0   )  ( x0   )   0
 0
 ( x) x    ( x) x    ( x) is continuous at x0 .
0
0
Have a fun!
Created by Tianxin Yang, College
of Precision Instruments and Optoelectronic Engineering, Tianjin University
Page 18 of 18