Download Lecture 3

Schrödinger's Equation
In the preceding lectures we found that we can some how calculate the wave
function or the probability Ψ(r,t) at all points at each instant of time then the
statistical moments such as average value, variance, standard deviation of all
other physically observable quantities such as position, momentum, angular
momentum, energy etc. can be calculated.
•The equation that allowed us to do that is the famous ScrÖdinger equation and
has the form

ˆ
i
H
t
•With the normalization condition on the wave function

  (r , t ) d  1.
2
•It is important to note first of all the above equation is a proposition or
postulate of Quantum Mechanics and thus cannot be proved.
•But its validity can be tested by comparing the results obtained from this
equations with various experimental situations.
•The operator H is the hamiltonian or the total energy operator and is the sum of
the kinetic energy operator and the potential energy operator. As explained
earlier this can be done by writing the classical expression of the Hamiltonian and
then replacing the position and momentum variables by their operator.
2



Hˆ 
 2  V (r )
2m
Remarks
•
It is important to note that the time derivative iħ∂/∂t is not associated as an
operator with any dynamical variable such as energy even though the Sch.
Equation suggest an equivalence between this differential operator and the
Hamiltonian and energy operator, particularly when the energy of the system
is conserved. All the cases we study actually correspond to such situation.
•
The preceding equation just tells us how the time evolution of the probability
amplitude takes place so that we can calculate the probability amplitude at any
time if the probability amplitude at a given time is known.
•
Also sometimes you come across the phrase time energy uncertainty principle.
This also has a different meaning as compared to position momentum
uncertainty principle. It implies that if we need to measure a quantum
mechanical energy level accurately then we need to measure it for infinite
time. Since that is not possible any energy level is determined only within an
error bar. We shall not study this aspect in this course any further.
Stationary Sch. Equation
•
The time dependent Schrödinger equation forces us to find out the
eigenvalue equation for the Hamiltonian operator. If we know the solutions
of the eigenvalue equation of the Hamiltonian operator then from these we
can calculate the wavefunction at any later time.
•
The eigenvalue equation for the energy or the Hamiltonian operator is also
known as time independent or the stationary Schrödinger equation. It can be
written as
•
Hn  Enn
The states Φn are known as energy eigenstates. If we remember the
definition of the eigenstates, these wavefunctions have definite energy and
everytime you make a measurement of the energy on an electron or any
other wavefunction given by such an wavefunction it yields same value of the
energy. Thus such states are very important.
Stationary states
•
Let us consider any such eigenstates with the energy eigenvalue En and
substitute this in the time dependent Scrödinger equation. We get
Thus if a wavefunction has the above iE form
of time dependence it’s modulus
t
i
square which gives
n
n
 Enn  n (t )  n (0)e 
t probability density
the
does not change in time. Thus
the electronic state ( or state of any other quantum mechanical object) is
known as the stationary state.
•
The formal way of obtaining the above stationary solution is to use the
method of separation valuable.
•
In this method we notice that in the time dependent Sch. equation the left
side is operator is given by a time derivative where as the Hamiltonian
operator is entirely dependent on spatial co-ordinate. Thus both of them
can be equated to a constant E and the solution can be factored in a
spatially dependent part and an entirely time dependent part.
Probability is conserved
• Since the total probability has to be conserved over entire space there
should be a continuity equation like one in electrodynamics due to charge
conservation.
•To get this multiply the Schrödinger equation by the complex conjugate
of the wave function,
   2 2


 * H   * i
 * 
 V 
t
 2m

And consider the complex conjugate of this equation.
   2 2

 *
H *   (i
)  
 V  *
t
 2m

If we subtract the complex conjugated equation from the original
Probability current
•We obtain i  ( * )   * 1   2 2   1   2 2  *
t
2m
2m
2
 * 
We add and subtract
2m
hand side
to the right
2
 ( * )
1


i
 *
  2 2  
( *  )
t
2m
2m
2
1



  2 2  * 
( *  )
2m
2m
Which can be written as
 ( * )
2
2
i

( *  ) 
(   *)
t
2m
2m
Probability Conserved
•We now define
   * ,
as the probability density
and




j i
(  *  * )
2m
• the probability current as
•Then our equation becomes
 

   j
t
This is the same equation of continuity as we had in the
case of charges and shows that the probability density is
locally conserved, just like the charge density is.
So if probability increases somewhere it is because
probability flows in from somewhere else.
General properties of the
wave function
•Because of the continuity equation we can give some
general requirements that wave functions must satisfy.
The only solutions of the Schrödinger equation of interest to
us are those for which,
1. The wave function is continuous.
2. The first derivatives of the wave function are continuous.
3. The wave function vanishes at spatial infinity.
We require the last as particles infinitely far away are of no
interest to us.
Free particle Solutions
•To get some idea of what all this means, we first solve the
Schrödinger equation for a free particle.



(
r
,t) 1



H (r , t )  i

  2 2  (r , t )
t
2m
•We consider first the case of one space dimension, say x.
 ( x, t ) 1  2  2 

H (r , t )  i


 ( x, t )
2

t
2m 
x 
•There is a standard way to solve such linear differential
equations in several variables which I give below. It is called
the method of separation of variables.
Separation of variables
•We try to find a solution of the form
 ( x, t )  ( x) (t ).
•Substituting this in the free particle equation, we obtain
(t ) (t )  2  2 
H (( x)(t ))  i( x)

  2  ( x )
t
2m 
x 
•We divide this equation by
 ( x)  (t )
 2 2 
1 (t )
1
H  i

( x)
 
2
(t ) t
2m( x) 
x 
•As the middle term is a function of t alone, while the last
is
Separation of variables
•A function of x alone and the two variables are
independent. The equation will be satisfied for all times if
and only if the terms on each side of the equals sign is a
constant, which we call E.
•Such a constant which we get by separating the
dependence of a function in several variables into a
product is called a separation constant. We can use this
separation constant to completely solve the problem of
time dependent Sch. equation
H  E ,

 2 2
H
  V (r )
2m
Free particle solution
•Comparing this with the eigenvalue equation we had
defined last time, and remembering that the operator H
represents the observation of the energy of the system,
then E is the eigenvalue of this operator or in other words
the energy we will observe for this particle.
•Now consider the other two parts of the equation.
Implies
i
t
d
 i t  Et
t
dt
t (t )  t 0e
i
Et

Free particle solution
•Here
is a constant of integration. Similarly we have
2
d

2

 2mE
2
dx
•This is the equation of a simple harmonic oscillator and has
t 0
the solution
i
( x)  Ae
px

 Be
i
px

,
p  2mE
•Combining all these results, we find that
 ( x, t )  Ce
i
px Et

 De
i
px Et

So the free particle can be represented by a plane wave
travelling in the +x direction or along the negative x.
Free particle solution
•The constants of integration C and D also have a physical
meaning.
C
2
C D
2
2
•
•Is the probability that the particle when measured will be
found to be travelling in the +x direction. With a similar
interpretation in the case of D.
Momentum conservation
•We now consider what at first sight seems to be an
unrelated problem.
•Consider the commutator of [ p, H ]
p2
p2
p2
[ p, ]  p ( )  ( ) p  0
2m
2m
2m
•Next consider the action of the momentum operator on our
eigenfunction for the free particle  . This corresponds to
the physical process of measuring the momentum of the free
particle.
Momentum eigenvalue
i
d
d
(i ) ( x, t )  (i )(Ce
dx
dx
 ( pCe
i
px Et

 pDe
i
px Et

px Et

 De
i
px Et

)
)
•So the two waves are separately eigenfunctions of
momentum also! One corresponding to positive momentum
and the other to negative momentum –p.
•Thus we obtain the interesting result that if two physical
operators commute with each other, they have
simultaneous eigenvalues which can be measured together.