Lecture 3 PHL110 QM

8/13/2019 Lecture 3 PHL110 QM

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Schrödinger's EquationIn the preceding lectures we found that we can some how calculate the wavefunction 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 allother physically observable quantities such as position, momentum, angularmomentum, energy etc. can be calculated.•The equation that allowed us to do that is the famous ScrÖdinger equation andhas the form

•With the normalization condition on the wave function

•It is important to note first of all the above equation is a proposition orpostulate of Quantum Mechanics and thus cannot be proved.•But its validity can be tested by comparing the results obtained from thisequations with various experimental situations.•The operator H is the hamiltonian or the total energy operator and is the sum ofthe kinetic energy operator and the potential energy operator. As explainedearlier this can be done by writing the classical expression of the Hamiltonian andthen replacing the position and momentum variables by their operator.

.1),( 2

d t r

H

t

i ˆ

)(2

ˆ 22r V

m H



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 momentumuncertainty 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

• 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.

nnn E H



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 form of time dependence it’s modulus

square which gives the probability density 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 leftside 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.

t iE

nnnnn

n

et E t

i

)0()(



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,

And consider the complex conjugate of this equation.

If we subtract the complex conjugated equation from the original

V

mt i H

2***

22

*2

)*(*

22

V

mt i H



Probability current

•

We obtain

We add and subtract to the right

hand side

Which can be written as

*2

1

2

1*

)*( 2222

mmt i

*2

2

m

)*(2*2

1

)*(22

1*

)*(

222

222

mm

mmt i

*)(2

)*(2

)*(

22

mmt i





General properties of the

wave function

•

Because of the continuity equation we can give somegeneral 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 theSchrödinger equation for a free particle.

•We consider first the case of one space dimension, say 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.

),(2

1),(),( 22 t r

mt

t r it r H

),(2

1),(),(

2

22 t x xmt

t xit r H



Separation of variables

•We try to find a solution of the form

•Substituting this in the free particle equation, we obtain

•We divide this equation by

•As the middle term is a function of t alone, while the last

is

).()(),( t xt x

)(2

)()()())()((

2

22

x xm

t

t

t xit x H

)()( t x

)()(2

1)(

)(

12

22

x x xmt

t

t i H



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 oftime dependent Sch. equation

)(2

,

22

r V m H

E H



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.

Impliest

t t E dt

d i

t i

Et i

t t et

0)(




•

Here is a constant of integration. Similarly we have

•This is the equation of a simple harmonic oscillator and has

the solution

•Combining all these results, we find that

So the free particle can be represented by a plane wave

travelling in the + x direction or along the negative x.

0

t

mE dx

d 2

2

22

mE p Be Ae x pxi pxi

2,)(

Et pxi

Et pxi

DeCet x

),(




•The constants of integration C and D also have a physical

meaning.

•

•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.

22

2

DC

C



Momentum conservation

•

We now consider what at first sight seems to be anunrelated problem.

•Consider the commutator of

• 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.

],[ H p

0)2

()2

(]2

,[222

pm p

m p p

m p p



Momentum eigenvalue

•So the two waves are separately eigenfunctions ofmomentum also! One corresponding to positive momentum

and the other to negative momentum –p.

•

Thus we obtain the interesting result that if two physicaloperators commute with each other, they have

simultaneous eigenvalues which can be measured together.

)(

))((),()(

Et pxi

Et pxi

Et pxi

Et pxi

pDe pCe

DeCedxd it x

dxd i

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Lecture 3 PHL110 QM