QUANTUM MECHANICS

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Course outline…

Quantum Mechanics: Wave equation, Time dependent Schrodinger equation, Linearity & superposition, Expectation values, Observables as operators, Stationary states and time evolution of stationary states, Eigenvalues & Eigenfunctions, Boundary conditions on wave function, Application of SE (Particle in a box, Potential barrier and step, one dimensional harmonic oscillator)

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References:A.Beiser, Concept of Modern Physics (or Perspective of Modern Physics), Tata-McGraw Hill, 2005.D. J. Griffith, Introduction to Quantum Mechanics, Pearson, 2007.

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Quantum Mechanics

“everything” about the system!

The methods of Quantum Mechanics consist in finding the wave function associated with a particle or a system

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HOWEVER

Ψ(x,t) is determined by Ψ(x, t = 0)

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Since the wavefunction, Ψ(x,t), describes a particle, its evolution in time under the action of the wave equation describes the future history of the particle

Thus, instead of the coordinate and velocity at t = 0 we want to know the wave function at t = 0

Thus uncertainty is built in from the beginning

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EM Waves in Empty SpaceMaxwell’s equations:

partial derivatives for the fields in empty space:2 2 2 2

2 2 2 2o o o oE E B Bμ ε and μ εx t x t

E = Emax sin (kx – ωt), B = Bmax sin (kx – ωt)

simplest solution: a sinusoidal wave:

wave number: k = 2π/ λ (λ is the wavelength)angular frequency: ω = 2πƒ (ƒ =1/t is the wave frequency)

smcv /1099792.21 8

00

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Classical Wave Operators

For a traveling wave solutions are the eigenvectors, sin(kx-t), and eigenvalues of the operator on the left hand side are -k2

For a standing wave between two reflecting mirros separated by a distance a, the eigenvalues are, -kn

2 = n22/a2

2

2

22

2 ),(1),(t

txEvx

txE

Consider the wave equation for light

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Any measurement of the observable a corresponds to operator Â, the only values that will ever be observed are the eigenvalues of Â, which satisfy the eigenvalue equation

ˆ , ,A x t A x t

*

*

ˆ, ,

, ,

x t A x t dxa

x t x t dx

If a system is in a state described by a wave function (x,t), then the average value of the observable a (measured once on many identical systems) is given by

The wave function of a system evolves in time: time-dependent Schrödinger equation:

2

2

,,

2x t

i U x x tt m x

Foundations of Quantum Mechanics

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Conditions on The must:

1. be a continuous and single-valued function of all x and t (the probability density must be uniquely defined)

2. have a continuous first derivative (the exception - points where the potential is infinite)

3. Have a finite normalization integral (so we can define a normalized probability)

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In 1D:

Calculate the normalization integral

Re-scale the wave function as

2( , ) dN x t x

- a wavefunction which obeys this condition is said to be normalized

The probability of finding a particle somewhere in space must be unity, thus the normalization condition:

2 3, 1all space

r t d r

2, 1x t dx

Suppose we have a solution to the Sch. Eq. that is not normalized. The recipe for normalization:

This procedure works because any solution of the S.Eq. being multiplied by a constant remains a solution: the S..Eq. is linear and homogeneous.

1' , ,r t r tN

Normalization

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Physical Quantity Operators

symbol actual operation

Momentum

Total Energy

Coordinate

Potential Energy

Quantum Mechanical Operators

)()(ˆ)(

ˆ

ˆ

ˆ

xUxUxU

xxxt

iEE

xipp xx

1111

Wave Function of Free Particle

For classical waves:]sin[],cos[ tkxAtkxA

Since the de Broglie expression is true for any particle, we assume that free particles can be described by a traveling wave, i.e. the wavefunction of a free particle is a traveling wave

However, these functions are not eigen functions of the momentum operator, with them we do not find

But let’s try operating on the following wave function with ,

xx pphhkxi

p

22

ˆ

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)exp(]sin[]cos[),( tikxAtkxitkxAtx

),(),(ˆ )()()( txkkAeikAei

Aexi

txp tkxitkxitkxix

Get same result of course if operate on

Similarly can operate on )exp(),( tikxAtx

tiE

ˆ

txEAeAeiiAet

itxE tkxitkxitkxi ,)(,ˆ )()()(

This wave function is an eigenfunction of both momentum and energy

with

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Expectation Values

Only average values of physical quantities can be determined (can’t determine value of a quantity at a

point)These average values are called Expectation Values

These are values of physical quantities that quantum mechanics predicts and which, from experimental point of

view, are averages of multiple measurements

Example, [expected] position of the particle

1)( with,)(

dxxPdxxxPx

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Expectation ValuesSince P(r,t)dV=|Ψ(r,t)|2dV, we have a way to

calculate expectation values if the wavefunction for the system (or particle) is known

dxtxxtxx

txtxtxdxtxxdxtxxPx

),(),(

),(),(),( since ,),(),(

*

*22

dxtxWtxW ),(ˆ),(*

In general for a Physical Quantity WBelow Ŵ is an operator acting on

wavefunction Ψ(r,t)

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Expectation Value for Momentum of a Free Particle

Generally

dxxxxip

dxxx

ixdxxpxp

)()(

)()()(ˆ)(

*

**

pkdxAeAekdxAeiki

Aep

dxAexi

Aep

A

dxAeAedxxAex

ikxikxikxikx

ikxikx

ikxikxikx

**

*

*2

n integratio of limits as 0 where

,1)( with )(

Free Particle

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Properties of the Wavefunction and its First Derivative

1. must be finite for all x2. must be single-valued

for all x3. must be continuous for

all x

dxxx

ixpx )()(*

dxxxUxxU )()()()( *

dxtxt

itxE ),(),(*

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Schrödinger Equation

xExxUxx

m

)(

2 2

22

xt

ixxUxx

m

)(2 2

22ExU

mp

)(2

2

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Schrödinger developed the wave equation which can be solved to find the wavefunction by translating the equation for energy of classical physics into the language of waves

For fixed energy, we obtain the time-independent Schrödinger equation, which describes stationary states

Energy of such states does not change with time ψn(x) is an eigen function or eigen state U is a potential function representing the particle interaction with the environment

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Stationary state

Assume V is independent of t , use separation of variables

Deduce from equation (2.1) , then

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...(2.2)

time-independent Schrödinger equation

Properties of (i) Linearity and superposition Wave functions add, not the probabilities

Linearity: An important properties of Schrodinger equation: it is linear in the , the equation has terms that contain and its derivatives but no terms independent of or that involve higher powers of or its derivatives.

Superposition: If 1 and 2 are two solutions, = a11 + a22 is also a solution; 1 and 2 obey the superposition principle.

Interference effects can occur for wave functions just as they can for light, sound, etc

Superposition's to the diffraction of an electron beam:

for every expectation value is constant in time Probability is independent of time

Slit 1 is open: probability density: P1 = I1I2 = 1* 1

Slit 2 is open: probability density: P2 = I2I2 = 2* 2

Both open, probability density at screen: P = II2 = I1+ 2I2 = (1* + 2*)( 1+ 2)

= 1* 1 + 2* 2 + 1* 2 + 2* 1

= P1+ P2+ 1* 2 + 2* 1

(ii) Stationary state Responsible for oscillations of the e- intensity at screen

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(ii) Definite total energy Classical mechanics : total energy is Hamiltonian

Quantum mechanics : corresponding Hamiltonian operator

thus equation (2.2)

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Applications of Quantum Mechanics

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1. Particle in a box/ infinitely potential well

2. Finite potential well

3. Potential barrier(tunnel effect)

4. Simple Harmonic Oscillator

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Particle in a box with “Infinite Hard walls”

Since the walls are impenetrable, there is zero probability of finding

the particle outside the box. Zero probability means:

ψ(x) = 0, for x 0 and x L

The wave function must also be 0 at the walls (x = 0 and x = L),

since the wavefunction must be continuous

Mathematically, ψ(0) = 0 and ψ(L) = 0

Boundary conditions and normalization determines + x 0

U(x) = 0 0 x L

+ x L

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We can re-write it as

xExx

m

2

22

2

22

22

2

22

2

2

2

mEk

xkxx

xmExx

For 0 < x < L, where U(x) = 0, the Schrödinger equation can be expressed in the form

Schrödinger Equation xExxUxx

m

)(

2 2

22

for Particle in a Box

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xkxx 2

2

2

The most general solution to this differential equation is:

ψ(x) = A sin kx + B cos kxA and B are constants which are determined from the properties of the ψ as well as boundary and normalization conditions

1. Sin(x) and Cos(x) are finite and single-valued functions

2. Boundary Condition: ψ(0) = ψ(L) = 0• ψ(0) = A sin(k0) + B cos(k0) = 0 B = 0

ψ(x) = A sin(kx)

• ψ(L) = A sin(kL) = 0 sin(kL) = 0 kL = nπ, n = ±1, ±2…

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22

22

2

2222

22

822

)(

2n

mLhn

mLm

nL

mkE n

n

...3,2,1 nnL

kn

Particle in a Box

Energy Levels:

• The allowed wave functions are given by

x

Ln A(x) ψnsin

• The normalized wave function:

x

Ln

L (x) ψn

sin2

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x

Ln

L (x) ψn

sin2 2(x)ψn

01

2

22

02

02

2

22

, )1( 2

,2

EEenergyn stategroundmL

EwithnEnmL

En

1ψ

2ψ

3ψ

4ψ

21ψ

22ψ

23ψ

24ψ

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Ex-1: e- in a 10nm wide Well with infinite barriers. Calculate E0 for L = 10 nm

Ex-2: Assume that a photon is absorbed, and the electron is transferred from the ground state (n = 1) to the second excited state (n = 3). What was the wavelengths of the photon?

2

22

012

0 2 where,

mLEEnEEn

eV 10 meV 1

meV 753eV 00375.0J 106

)1010(101.92)1005.1(14.3

3-

220

2931

2342

0

.E

E

30

eV 00375.001 EEEground

eV 0.0338eV 00375.093 is state excited Third

203

3

EEE

μm 41nm 41333030

1240eV 03000037500.0338)( 13

.λ

..EEh

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Average Momentum of Particle in a Box (Infinite Potential Well) problem

Note: the right hand side is either 0 or imaginary, but momentum cannot be imaginary so it must be zero

0)cos()sin(2

sin2

]sin2[

)()(

0

0

0

**

L

L

L

x

dxkxkxkiL

dxx

kxL

ikx

L

dxxxi

xp

02 xp

But

Why ???

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SUMMARY

Features Classical behaviors Quantum behaviors

1. Allowed energy levels

2. Minimum energy

3. Position probability

A classical particle have any energy

The minimum energy of the particle is zero

It has a value at all points within the well

The particle can have discrete energy values given by

The minimum energy of the particle is

The probability of finding the quantum mechanical particle at position x depends on In(x)I2 and hence has different points and for different states.

Classical and quantum behaviour of a particle confined in one dimensional Box

...3,2,1;2

22

22

nnmL

En

2

22

1 2mLE

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Application No. 2 Finite Potential Well

♣ The potential energy is zero (U(x) = 0) when the particle is 0 < x < L (Region II)

We also assume that energy of the particle, E, is less than the “height” of the barrier, i.e. E < U

♣ The energy has a finite value (U(x) = U) outside this region, i.e. for x < 0 and x > L (Regions I and III)

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Finite Potential Well

Schrödinger Equation

I. x < 0; U(x) = U

II. 0 < x < L; U(x) = 0

III. x > L; U(x) = 0

IIII E

dxd

m

2

22

2

III EU

dxd

m

2

22

2

xExxUxx

m

)(

2 2

22

IIIIIIIII EU

dxd

m

2

22

2

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Finite Potential Well: Region II

U(x) = 0This is the same situation as previously for infinite potential well

The allowed wave functions are sinusoidal

The general solution of SE is

ψII(x) = F sin kx + G cos kx:F and G are constants

♣ Boundary conditions, however, no longer require that ψ(x) be zero at the sides of the well

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Finite Potential Well: Regions I and III

EUdxd

m 2

22

2

222

22

2 )(2 where,)(2

EUmCCEUmdxd

mUC 2

The Schrödinger equation for these regions is

It can be re-written as

The general solution of this equation is

ψ(x) = AeCx + Be-Cx

A and B are constantsNote (E-U) is the negative of kinetic energy, -Ek In region II, C is imaginary and so have sinusoidal solutions we found

In both regions I and III, ψ(x) is exponential

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Requires that wavefunction, ψ(x) = AeCx + Be-Cx

not diverge as x ∞

So in region I, B = 0, and ψI(x) = AeCx

to avoid an infinite value for ψ(x) for large negative values of x

Finite Potential Well: Regions I and III

In region III, A = 0, and ψIII(x) = Be-Cx

to avoid an infinite value for ψ(x) for large positive values of x

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Finite Potential Well

This, together with the normalization condition, determines the amplitudes of the wavefunction and the constants in the exponential term.

This determines the allowed energies of the particle

• The wavefunction and its derivative must be single-valued for all x– There are two points at which wavefunction is given

by two different functions: x = 0 and x = L

Ldx

dLdx

dLL

dxd

dxd

IIIII

IIIII

III

III

)()(

00

)0()0(

Thus, we equate the two expressions for the and its derivative at x = 0, L

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Finite Potential WellGraphical Results for ψ (x)

Outside the potential well, classical physics forbids the presence of the particleQuantum mechanics shows the wave function decays exponentially to zero

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2

8n

mLhEn

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Finite Potential WellGraphical Results for Probability Density,

| ψ (x) |2

The functions are smooth at the boundaries

Outside the box, the probability of finding the particle decreases exponentially, but it is not zero!

http://phys.educ.ksu.edu/vqm/html/probillustrator.html

The probability densities for the lowest three states are shown

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TunnelingThe potential energy has a constant value U in the region of width L and zero in all other regionsThis a called a barrier

U is the called the barrier height. Classically, the particle is reflected by the barrier

Regions II and III would be forbidden• According to quantum mechanics, all regions are

accessible to the particle– The probability of the particle being in a classically

forbidden region is low, but not zero– Amplitude of the wave is reduced in the barrier – A fraction of the beam penetrates the barrier

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Simple Harmonic Oscillator(SHO)

solution

Classical treatment :

Harmonic Motion:Vibrates about an equilibrium configuration

Condition: presence of restoring force that acts to return the system to its equilibrium position when it is disturbed

For SHM:

Potential energy V is related to F :

Simple Harmonic Oscillator(SHO)Quantum treatment :

WHY TO STUDY:This approach indentifies several problems:1.diatomic molecule2.an atom in a crystal lattice etc3.explain blackbody radiation;

Planck postulated that the energy of a SHO is quantized.(In his model vibrating charges act as simple harmonic oscillators and emit EM radiation)

Let’s write down the Schrödinger Equation for SHO

22)(

222 xmkxxU

mk

For SHO the potential energy is

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xExxmxx

m

22

22

2

22Time independent Schrödinger Equation for SHO in 1D

Simple Harmonic Oscillator(SHO)

Solutions: To obtain and E

Algebraic Method

Exmdxdi

m

2

2

21

........................(1)

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rewrite equation (1) by ladder operator :

compare equation(1)

similarly

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Discussions

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Simple Harmonic Oscillator(SHO)There must exist a min state with

and from

so the ladder of stationary states can illustrate :

21nEn

n = 0, 1, 2,….

2

2xm

nnn eaAx

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Energy of SHO from the Schrödinger equation

hnhE21

The zero point energy ½hν is required by the Heisenberg uncertainty relationship

The term of ½hν is important for understanding of some physical phenomena

For example, this qualitative explains why helium does not become solid under normal conditions:

the “zero point vibration” energy is higher than the “melting energy” of helium

Force between two metal plates

Term ½hν tells us that quantum SHO always oscillates. These are called zero point vibrations

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• Decaying wavefunction tunnels into classically forbidden region• Spatial average for high energy wavefunction gives classical result: another example of the CORRESPONDENCE PRINCIPLE