Module 9 – Schrodinger Equation
I.
Wave Function
A.
Importance

A quantum system is described by a wave function, Ψ( r , t) , that contains all
information (linear momentum, angular momentum, energy, etc.) about a the
system.
In Classical physics, the position vector and its derivatives are the key to
describing a system. In Q.M. problems, the wave function replaces the position
function as the central focus.
B.
Properties
To represent a physical system, the wave function must have the following
properties:
1.
2.
3.
4.
The wave function must be continuous.
The wave function must be finite.
The wave function must be single valued for a specific location in space
and time.
For any finite potential, the first spatial derivatives of the wave function
must be continuous. (i.e.
5.
Ψ
is continuous)
x
The square of the modulus of the wave function must be normalized.

(i.e.
 Ψ Ψ dx 1)


C.
Probability
The telltale experimental evidence of waves is the property of interference. By
considering the results of electron scattering experiments like the demonstration
in class, Max Born realized that the interaction between a viewing screen and an
individual electron showed a particle nature (localized spot). The interference
pattern was formed due to the distribution of the spots when a large number of
electrons were scattered. Thus, he hypothesized that the wave nature is manifested
in the probability density function that is equal to the square of the wave function.
This is known as the Born Postulate and it forms the basis of the standard
interpretation of Quantum Mechanics (Copenhagen Interpretation).
Mathematically, the probability of finding a particle between x and x+dx is given
by

Ρ(x, t)   (x, t) (x, t) dx  (x, t) dx .
2
It should be noted that the wave functions for physical systems are complex.
However, the probability density is the square of the modulus of the complex
function (i.e. the square of the magnitude of the vector in the complex plane)
which is a real number. If you are rusty on complex numbers, I suggest that you
consult Arfkin.
II.
Time Dependent Schrodinger Equation
Erwin Schrodinger objected to deBroglie’s wave hypothesis and the hodge-podge
nature of old quantum mechanics. Schrodinger argued that is a QM system has
wave properties then it must obey a wave equation. Eventually Schrodinger was
able to find an equation which was obeyed by non-relativistic particles.



2 2 
Ψ r , t 

 Ψr , t  U r , t Ψ r , t  i
2m
t
“3-D”
 2  2 Ψx, t 
Ψx, t 

 Ux, t Ψx, t   i
2
2m x
t
“1-D”
You can’t prove the Time Dependent Schrodinger Equation any more than you
can prove Newton’s 2nd Law or Maxwell’s Equations. The proof is that the
equation correctly predicts the results of experiments. We can show that the
equation makes physical sense.
A.
Requirements
To develop a suitable wave equation, Schrodinger needed to ensure the following
things:
1.
2.
3.
4.
The solutions to the equation are waves.
The equation is consistent with deBroglie’s relationship.
The equation is consistent with Einstein’s relation.
The equation is consistent with conservation of energy.
B.
Momentum Operator - p̂
Since we want a wave equation, we will start by assuming that our wave function
i  kxω t 
is a plane wave (i.e. Ψ(x, t)  e
). Any wave equation must involve partial
derivatives with respect to space and time. We therefore start by finding out what happens to our
wave function when we take a spatial derivative.
Ψ  e i kx  ω t  

x
x
Ψ i kx  ω t  ikx  ω t  
e
x
x
Ψ
i k Ψ
x
i
Ψ
 kΨ
x
We now multiply both sides by  and apply deBroglie’s relationship.
 k Ψ  i
p̂ Ψ   i 
Ψ
x
Ψ
x
Our equation indicates that linear momentum, p̂ , is acting as an operator instead
of a number. In QM every physical quantity is associated an operator which acts
upon the wave function to obtain the individual values (eigenvalues) of the
quantity which can be measured.
C.
Energy Operator - Ĥ
We now look at the temporal derivative of our wave function.
Ψ  e i kx  ω t  

t
t
Ψ i kx  ω t   ikx  ω t  
e
t
t
Ψ
  i ωΨ
t
i
Ψ
 ωΨ
t
We now multiply both sides by  and apply Einstein’s relation.
 ωΨ  i
Ĥ Ψ  i 
Ψ
t
Ψ
t
The energy operator is called the Hamiltonian. The formulation for old quantum
mechanics came out of an alternative formulation of Classical Mechanics called
Hamiltonian Mechanics which was based upon scalar quantities instead of vector
quantities as in Newton’s Mechanics. You will study Hamiltonian Mechanics in your
Advanced Classical Mechanics course.
D.
Conservation of Energy
The total energy of a classical system is the sum of the system’s kinetic and
potential energy. Thus, we have that
p̂ 2
E
 Ux, t 
2m
We now replace each of these quantities with their corresponding operators which
we have developed in the preceding sections and apply it to the wave function.
p̂ 2
Ψx, t  Ux, t Ψx, t  Ĥ Ψx, t 
2m
 
 

 - i  - i 
x 
x 


Ψx, t   Ux, t Ψx, t   i Ψx, t 
2m
t
 2  2 Ψx, t 
Ψx, t 






U
x,
t
Ψ
x,
t

i

2 m x 2
t
The result is the Time Dependent Schrodinger Equation which is consistent with
all of our requirements because we used each one of them in developing the
equation. Again, I repeat that this isn’t a proof, but does show that the
Schrodinger Equation makes sense. Also, because the equation treats space and
time different it is not applicable to relativistic problems (p2/2m isn’t kinetic
energy in a relativistic problem).
III.
Time Independent Schrodinger Equation
A.
Using Separation of Variables
When the Hamiltonian is not a function of time, it is possible to separate the
time and spatial parts of the wave function!!
The Time Dependent Schrödinger Equation can then be attacked using a
differential equation solution technique called “Separation of Variables.”
The first step is to write the wave function as a product of a function which only
depends on space and a second function that only depends on time. In the case of
a 1-dimensional spatial problem, we have
 x, t   Φ x  T t 
We now substitute the equation into our Time Dependent Schrödinger Equation.
i

 x, t   Ĥ x   x, t 
t
i   x 
 T t 
 Tt  Ĥ x Φx 
t
It is important to remember that the Hamiltonian includes partial derivates and
other operators that operate on the spatial variable x. Thus, a function of t can be
moved to the left of the operator but not a function of x. A similar argument holds
for the left-hand side of the equation where only functions of the spatial variable
can be pulled outside the partial derivative with respect to time.
i   Tt 
1

Ĥ x Φx 
Tt  t
 x 
The left hand side of the equation contains only the variable t while the right hand
side of the equation contains only the variable x. This equation must be true for
any value of t or x. Since we could arbitrarily choose to just vary t or to just vary
x, the only way that this equation will be valid is if the equation is equal to a
constant which we will call E.
i   Tt 
1

Ĥ x Φx   E
Tt  t
x 
We now have broken the Time Dependent Schrödinger Equation containing
both x and t into two equations each involving only one variable. With some
simple algebra, we have
d T t 
iE
  T t 
dt

Time Equation
Ĥ x  x   E Φx 
Time Independent Schrödinger Equation
The Time Independent Schrödinger Equation is an eigenvalue equation with
the constant E being the energy eigenvalue associated with Φx  the energy
eigenfunction. Thus, we see that solving the Time Independent Schrödinger
Equation is to find the energy vales and energy state of a quantum system!! In
general, there may be several eigenfunctions (solutions) that fit the equation.
Thus, the most general solution will be a sum of all of the individual
eigenfunctions.
B.
Solution to the Time Dependence Equation
What about the solution to the Time Equation? Using Einstein’s energy
relationship, we have that
d T t 
  i ω T t 
dt
We have seen this particular equation and been required to know its solution ever
since Physics 1224. The solution is
Tt  e  i ω t
It is a vector in the complex plane that rotates at an angular frequency dependent
on the energy of the state. It is important to note that a wave function may be
composed a superposition (sum) of several different energy eigenfuctions. Each
of these eigenfuctions will evolve at a different angular frequency if they have
different energies.
C.
What have we gained by this exercise?
1.
The wave function for a quantum mechanical system who’s Hamiltonian
is not a function of time can always be written as a sum of energy
eingenfunctions
x, t 
 Φ x e
n
 i ωn t
n
2.
To find the energy eigenfunctions and energy eigenvalues for a quantum
mechanical system who’s Hamiltonian is not a function of time, you must
solve the simple Time Independent Schrödinger Equation.
Ĥ x Φx   E  x 

2 2
Φx  Ux Φx   E Φx 
2m  x 2
We obviously need to know the potential energy function for the
problem in order to solve the Time Independent Schrödinger
Equation just like you need to know the forces in a problem to solve
Newton’s II law.
II.
Solving the Time Independent Schrödinger Equation
A.
Since we will spend a great deal of time solving the Time Independent
Schrödinger Equation, it is a good idea to look at the equation in some detail. We
start by writing the Time Independent Schrödinger Equation as follows:
2
2mE - V(x) 
Φx   
Φx 
2
x
2
The term E - V(x) is the kinetic energy of the particle.
The term 2mE - V(x) is the linear momentum of the particle.
In classical physics, a particle can only exist in regions where the kinetic energy is
greater or equal to zero otherwise its linear momentum and hence velocity would
be imaginary. The region where the kinetic energy is negative is called the
forbidden region and the values of x where the kinetic energy and velocity
equal zero are called the classical turning points. In both classical and quantum
physics, the behavior of waves is different and dictated by the wave number.
According to the deBroglie hypothesis, the square of the wave number is the
2mE - V(x) 
term
.
2
If the wave number is real, then we have a traveling wave which represents a
moving particle. In the forbidden region, the wave number is imaginary!! In this
case, we have an evanescent wave whose amplitude decays exponentially. The
evanescent wave can’t travel to a detector since it has an imaginary linear
momentum, but its presence can be detected by detecting the passage of the wave
through a potential barrier. For instance a light ray can be sent into a prism at an
angle of incidence greater than the critical angle so that total internal reflection is
observed. However, if a second prism is brought sufficiently close to the first
prism, then a light ray will appear in the second prism even though no light
appears across the gap between the prisms (see Eisberg & Resnick)!! This wave
phenomenon is known as tunneling and it has important consequences for
radioactive decay, electronics, and nuclear reactions.
B.
Constant Wave Number
An important special case for solving differential equations is when the Time
Independent Schrodinger Equation either has a constant wave number for
the whole problem or when the problem can be broken into regions where
the wave number is constant.
1.
Linear Solution – k2 = 0
Our differential equation now becomes
2
Φx   0
 x2
This problem is easily solved by integrating the equation twice with respect to x
thereby obtaining the equation of a straight line.
Φx   m x  b
You encountered this result in College Algebra as well several areas of physics.
For instance, the general equation of motion for a particle traveling in the xdirection with no acceleration is obtained from Newton’s II Law
d2 x
0.
d t2
This gives us the well know result that
x  v t  xo .
In quantum mechanics, the linear solution usually fails to have the required
properties for physical systems and is rarely seen.
2.
Parabolic Solution – k2 > 0
This equation is also an old friend which we encountered in our work on
harmonic oscillators in both Phys1224 and Phys2424. Its solution is sine and
cosine functions from trigonometry. There are actually three standard ways to
write the solution to this problem:
a.
Single Trigonometry Function With Phase Angle
Φx   A Cos k x  θ
The constants A (amplitude) and  (phase angle) are a result of doing two
integrations. There values select specific solutions and are usually
determined by boundary conditions and normalization (in QM).
b.
Two Trigonometry Functions
An alternative form of the solution which is very useful for solving
bound problems is to write the solution as the sum of a sine and cosine
function.
Φx   A Cos k x   B Sin kx 
The amplitudes of the two trigonometry functions (A & B) are the two
constants. This form is extremely useful when boundary conditions require
one of the coefficients to be zero.
c.
Complex Exponential Forms
You should know that the sine and cosine functions are related to complex
exponentials through the relationships
e i  e i
Cos   
2
Sin   
e i  e  i
2i
Thus, we can also write our solution in terms of complex exponentials.
This is usually beneficial when dealing with a free particle that can be
represented by a plane wave or when we plan to perform a lot of
Calculus (phasor transforms, etc) since the exponential is its own
derivative. In this form or solution is
Φx   A e i kx  B e i kx
3.
Hyperbolic Solution – k2 < 0
In this case, the wave number is imaginary. Using the substitution,
2 = - k2, we can write the solution to the problem in one of two ways.
a.
Exponential Form
Φx   A e x  B e   x
This form is most useful in barrier problems or when one of the two
constant (A & B) must go to zero (i.e. a problem in which the solution
region contains either x = +∞ or x = - ∞.
b.
Hyperbolic Trigonometry Functions
Φx   A Cosh α x   B Sinh α x 
This form is most useful for bound systems where symmetry can force one
of the constants to zero (i.e. Cosh is an even function while Sinh is an odd
function). A common application is where the function is zero at x = 0 in
which case only the Sinh solution is allowed.