Physics-I

Dr. Anurag Srivastava

Web address: http://tiiciiitm.com/profanurag

Email: profanurag@gmail.com

Visit me: Room-110, Block-E, IIITM Campus

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• Electrodynamics: Maxwell’s equations: differential and integral forms, significance of Maxwell’s equations, displacement current and correction in Ampere’s law, electromagnetic wave propagation, transverse nature of EM waves, wave propagation in bounded system, applications.

• Quantum Physics: Dual nature of matter, de-Broglie Hypothesis, Heisenberg uncertainty principle and its applications, postulates of quantum mechanics, wave function & its physical significance, probability density, Schrodinger’s wave equation, eigen values & eigen functions, Applications of Schroedinger equation.

Syllabus

Erwin Rudolf Josef Alexander Schrödinger

Born: 12 Aug 1887 in Erdberg, Vienna, Austria

Died: 4 Jan 1961 in Vienna, Austria

Nobel Prize in Physics 1933

"for the discovery of new productive forms of atomic theory"

An equation for matter waves

2

2Try

t x

2 2

2i

2t m x

Seem to need an equation

that involves the first

derivative in time, but the

second derivative in space ( , ) is "wave function" associated with matter wavex t

,

i kx tx t e

As before try solution

So equation for matter waves in free space is

(free particle Schrödinger equation)

• Solve this equation to obtain y

• Tells us how y evolves or behaves

in a given potential

• Analogue of Newton‟s equation

in classical mechanics

Schrödinger’s Equation

Erwin Schrödinger (1887-1961)

• This was a plausibility argument, not a derivation. We believe the Schrödinger equation not because of this argument, but because its predictions agree with experiment.

• 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)

• The Schrödinger equation is a partial differential equation in x and t (like classical wave equation). Unlike the classical wave equation it is first order in time.

• 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).

Schrödinger’s Equation

Interpretation of |y|2

• The quantity |y|2 is interpreted as the probability that the

particle can be found at a particular point x and a

particular time t

• The act of measurement „collapses‟ the wave function

and turns it into a particle

Neils Bohr (1885-1962)

An equation for matter waves

2

( , )2

pE V x t

m

2 2

2i ( , )

2V x t

t m x

For particle in a potential V(x,t)

Suggests modification to Schrödinger equation:

Time-dependent Schrödinger equation

Total energy = KE + PE

Schrödinger

What about particles that are not free?

2 2

, ,2

kx t x t

my y

Has form (Total Energy)*(wavefunction) = (KE)*(wavefunction)

2 2

2i

2t m x

,i kx t

x t e

Substitute into free particle equation

2

2

pE

m

(Total Energy)*(wavefunction) = (KE+PE)*(wavefunction)

gives

The Schrödinger equation: notes

2 2

2i ( , )

2V x t

t m x

• This was a plausibility argument, not a derivation. We believe the Schrödinger

equation not because of this argument, but because its predictions agree with

experiment.

• 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)

• The Schrödinger equation is a partial differential equation in x and t (like

classical wave equation). Unlike the classical wave equation it is first order in time.

• 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).

• Note the +ve sign of i in the Schrödinger equation. This came from our looking

for plane waves of the form

We could equally well have looked for solutions of the form

Then we would have got a –ve sign.

This is a matter of convention (now very well established).

i te i te

The Hamiltonian operator

2 2

2i ( , )

2V x t

t m x

ˆi Ht

Can think of the RHS of the Schrödinger equation as a

differential operator that represents the energy of the

particle.

Hence there is an alternative

(shorthand) form for the time-

dependent Schrödinger equation:

This operator is called the Hamiltonian of

the particle, and usually given the symbol H2 2

2ˆ( , )

2

dV x t H

m dx

Kinetic

energy

operator

Potential

energy

operator

Hamiltonian is a linear differential operator.

Schrödinger equation is a linear homogeneous partial differential equation

Time-dependent

Schrödinger equation

Interpretation of the wave function

2 2

0( , ) ( , ) d

bb

xx a a

x t x x t x

2 * 2

( , )x t x

Ψ is a complex quantity, so how can it correspond to real physical

measurements on a system?

Remember photons: number of photons per unit volume is proportional to

the electromagnetic energy per unit volume, hence to square of

electromagnetic field strength.

Postulate (Born interpretation): probability of finding particle in a small

length δx at position x and time t is equal to

Note: |Ψ(x,t)|2 is the probability per unit length. It is

real as required for a probability distribution.

Total probability of finding particle

between positions a and b is

a b

|Ψ|2

x

δx

Born

An equation for matter waves:

the time-dependent Schrödinger equation

2 2

2 2 2

1

x v t

wave velocityv

Classical 1D wave

equation

e.g. waves on a string:

Can we use this to describe matter waves in free space?

( , )x t

x ( , ) wave displacementx t

Try solution ,

i kx tx t e

But this isn‟t correct! For free particles we know that

2

2

pE

m

DOUBLE-SLIT EXPERIMENT REVISITED

Detecting

screen

Incoming coherent

beam of particles

(or light)

sind

D

θ

y

1

2

Schrödinger equation is linear: solution with both slits open is 1 2

Observation is nonlinear 1 2

2 2 2 * *

1 2 2 1

Usual “particle” part Interference term

gives fringes

Normalization

Total probability for particle to be somewhere should always be one

Suppose we have a solution to the

Schrödinger equation that is not

normalized. Then we can

•Calculate the normalization integral

•Re-scale the wave function as

(This works because any solution to

the SE multiplied by a constant

remains a solution, because the SE

is LINEAR and HOMOGENEOUS)

Normalization condition 2( , ) d 1x t x

1( , ) ( , )x t x t

Ny y

2( , ) dN x t x

New wavefunction is normalized to 1

A wavefunction which obeys this

condition is said to be normalized

2 2( , ) ,

( , ) 0,

x t a x a x a

x t x a

Particle with un-normalized wavefunction

at some instant of time t

Normalizing a wavefunction - example

Conservation of probability

Total probability for particle to be somewhere should ALWAYS be one

2( , ) d constantx t x

If the Born interpretation of the wavefunction is correct then the normalization

integral must be independent of time (and can always be chosen to be 1 by

normalizing the wavefunction)

We can prove that this is true for physically relevant wavefunctions

using the Schrödinger equation. This is a very important check on the

consistency of the Born interpretation.

2 2

2i ( , )

2V x t

t m x

2( , ) d constantx t x

Boundary conditions for the wavefunction

The wavefunction must:

1. Be a continuous and single-valued

function of both x and t (in order that

the probability density is uniquely

defined)

Examples of unsuitable wavefunctions

Not single valued

Discontinuous

Gradient discontinuous

x

( )xy

( )xy

x

( )xy

x

2. Have a continuous first derivative

(except at points where the potential is infinite)

3. Have a finite normalization integral

(so we can define a normalized probability)

Time-independent Schrödinger equation

Suppose potential is independent of

time

2 2

2i ( )

2V x

t m x

LHS involves only

variation of Ψ with t

RHS involves only variation of Ψ

with x (i.e. Hamiltonian operator

does not depend on t)

Look for a separated solution ( , ) ( ) ( )x t x T ty

Substitute:

, ( )V x t V x

2 2

2( ) ( ) ( ) ( ) ( ) ( ) ( )

2x T t V x x T t i x T t

m x ty y y

2 2

2 2( ) ( ) ( )

dx T t T t

x dx

yy

2 2

2( )

2

d dTT V x T i

m dx dt

yy y

N.B. Total not partial

derivatives now

etc

2 2

2

1 1( )

2

d dTV x i

m dx T dt

y

y

Divide by ψT

LHS depends only on x, RHS depends only on t.

True for all x and t so both sides must be a constant, A (A = separation constant)

2 2

2( )

2

d dTT V x T i

m dx dt

yy y

1 dTi A

T dt

2 2

2

1( )

2

dV x A

m dx

y

y

So we have two equations, one for the time dependence of the wavefunction and

one for the space dependence. We also have to determine the separation constant.

This gives

/( ) iEtT t aeThis is like a wave i te

with /A . So A E .

1 dTi A

T dt

dT iAT

dt

/( ) iAtT t ae

1 dTi A

T dt

2 2

2

1( )

2

dV x A

m dx

y

y

SOLVING THE TIME EQUATION

• 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 V(x). But there is an implicit

dependence because the potential affects the possible values for the energy E.

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TIME INDEPENDENT SCHROEDINGER EQUATION

Erwin Schroedinger

Consider a particle of mass „m‟, moving with a velocity

„v‟ along + ve X-axis. Then the according to de Broglie

Hypothesis, the wave length of the wave associated

with the particle is given by

mv

h

A wave traveling along x-axis can be represented by the equation

xktieAtx ,

22

Where Ψ(x,t) is called wave function. The differential equation of matter wave in one

dimension is derived as

08

2

2

2

2

yy

VEh

m

dx

d

The above equation is called one-dimensional Schroedinger’s wave equation in one

dimension.In three dimensions the Schroedinger wave equation becomes

08

08

2

22

2

2

2

2

2

2

2

2

y

y

yyyy

VEh

m

VEh

m

zyx

2 2

2

d( )

2 dV x E

m x

yy y

With A = E, the space equation becomes:

This is the time-independent Schrödinger

equation

H Ey y

Solving the space equation = rest of course!

Time-independent Schrödinger equation

or

But probability

distribution is static

2 * / /

2*

, , ( ) ( )

( ) ( ) ( )

iEt iEtP x t x t x e x e

x x x

y y y

y y y

/( , ) ( ) ( ) ( ) iEtx t x T t x ey y Solution to full TDSE is

Even though the potential is independent of time the wavefunction still oscillates in time

For this reason a solution of the TISE is known as a stationary state

Notes

• In one space dimension, the time-independent Schrödinger equation is an ordinary differential equation (not a partial differential equation)

• The time-independent Schrödinger equation is an

eigenvalue equation for the Hamiltonian operator:

Operator × function = number × function

(Compare Matrix × vector = number × vector)

• We will consistently use uppercase Ψ(x,t) for the full wavefunction (TDSE), and lowercase ψ(x) for the spatial part of the wavefunction when time and space have been separated (TISE)

H Ey y

2 2

2

d( )

2 dV x E

m x

yy y

Energy of the electron

Energy is related to the Principle Quantum number, n.

This gives 3 of the 4 quantum numbers, the last one is the spin

quantum number, s, either +½ or – ½.

Wave

Functions: Probability to find

an electron

Energy of the electron

Electron Transitions give off Energy as

Light/Xrays

E=hc/

Red/Blue Clouds in Space

Zeeman Effect

Light Emission in Magnetic Field

Multi-electron Atoms

(Wolfgang) Pauli Principle Exclusion Principle

No 2 electrons with same quantum numbers!

Energy Eigenvalues

Eigen value and Eigen Function

Applications:

Operators in Quantum Mechanics