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Chapter 40 Quantum MechanicsApril 6,8 Wave functions and Schrödinger equation40.1 Wave functions and the one-dimensional Schrödinger equation

Quantum mechanics: Physical science studying the behavior of matter on the scale of atomic and subatomic levels.

A photon described by an electromagnetic wave:Probability per unit volume of finding the photon in a given region of space at an instant of time Square of the amplitude of the electromagnetic wave.

Interpretation of the wave function of a particle:

Wave function: A wave function describes the distribution of a particle in

space. The quantity is the probability that the particle can be found

within the volume dV around the point (x, y, z) at time t.

)(x,y,z,tΨdVx,y,z,tΨ

2)(

)(x,y,z,tΨ

particle. about theknown becan n that informatio theall containsfunction waveThe 4)

. inside particle thefinding of yProbabilit :)( 3)

density.y probabilit ,functionon distributiy Probabilit :)( )2

state. amplitude,y probabilit ,function Wave :)( 1)

:functions wave toNotes

2

*2

dVdVx,y,z,tΨ

ΨΨx,y,z,tΨ

x,y,z,tΨ

2

Wave packets:Wave packet: A wave packet is a wave that has a narrow distribution in space, so that it exhibits properties of a particle. A wave packet can be constructed by the sum of a large number of waves with a continuous distribution of similar wavelengths:

A wave packet has both the characteristics of a wave and of a particle.

dkekAtx tkxi )()(),(

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The uncertainty principle:

A broad distribution of A(k) results in a narrower wave packet.A short laser pulse must be white.

.2/12/ kxpx x

4

One dimensional Schrödinger equation: Erwin Schrödinger: (1887-1961)• Austrian physicist.• Famous for his contributions to

quantum mechanics, especially the Schrödinger equation.

• Nobel Prize in 1933.

A particle of mass m is confined to move along the x axis and interact with its environment through a potential energy function U(x). The wave function Y (x,t) satisfies

t

txitxxU

x

tx

m

),(

),()(),(

2 2

22

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Stationary states:Stationary state: A stationary state of a particle is a state that has a definite energy.The wave function of a stationary state can be written as a product of a time-independent wave function y (x) and a simple function of time:

/)()( iEtexx,t Notes to stationary states:1) Stationary states are of essential importance in quantum mechanics.2) A system can be in a state that is different from a stationary state and thus does not

have a definite energy. However, a wave function can always be decomposed into a combination of stationary wave functions.

3) At a stationary state the probability density function does not depend on time: 22/2

)()()( xexx,t iEt

Time-independent Schrödinger equation:

ExU

dx

d

m )(

2 2

22

A particle of mass m is confined to move along the x axis and interact with its environment through a potential energy function U(x). The total energy of the system is E, and the wave function of the system is y (x), then

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More about the time-independent Schrödinger equation:

• The first term in the Schrödinger equation represents the kinetic energy K of the particle multiplied by y , therefore K + U = E.

• If U(x) is known, one can solve the equation for y (x) and E for the allowed states. Some restrictions:1) y (x) must be continuous,2) y (x) 0 when x ±∞ (normalization condition), 3) dψ/dx must be continuous for finite values of U(x).

• Solutions of the Schrödinger equation may be very difficult.• The Schrödinger equation has been extremely successful in explaining the behavior of

atomic and nuclear systems.• When quantum mechanics is applied to macroscopic objects, the results agree with

classical physics.

ExU

dx

d

m )(

2 2

22

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Wave function for a free particle:

tiikxAtE

iikxAtx

Am

p

m

kE

mEk

ikxAikxAxEdx

d

m

expexpexp),(

.0 ly take temporarisLet'

.22

,2

expexp)( 2

11

2

222

212

22

Example 40.1Example 40.2Test 40.1

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Read: Ch40:1Homework: Ch40: 4,5,6,8 Due: April 17

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April 10 Particle in a box

40.2 Particle in a boxPotential well: An upward-facing region of a potential energy diagram. (opp. barrier).

Potential energy of a box:

Otherwise

0 0)(

LxxU

ExU

dx

d

m )(

2 2

22Schrödinger equation:

mE

kkmE

dx

d 2 ,

2 222

2

The general solution to this equation is

.cossin)( kxDkxCBeAex ikxikx

In the region x< 0 and x > L, where U = ∞, y (x)=0.In the region 0 < x < L, where U = 0, the Schrödinger equation is

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Applying boundary conditions to the general solution .cossin)( kxDkxCx

3,2,1 sin)( 0sin0)(

,00)0(

nx

L

nAx

L

nknkLkLCL

D

n

x

L

nCxn

sin)(

Energy levels:

,3,2,1 ,8

2

2

22

n

mL

hnE

L

nmEk n

2

22

8mL

hnEn

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Probability density:

,3,2,1 ,sin)( 222

nx

L

nCx

Normalization: The total probability of finding the particle somewhere in the universe must be 1.

.sin2

),( sin2

)(

21sin)(

/

222

tiEnexL

n

Ltxx

L

n

Lx

LCdxx

L

nCdxx

Uncertainty principle: For the state n =1,

.22

2

x

xx

px

Lkpkp

Lx

Example 40.3Example 40.4Test 40.2

1)(2

dxx

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Read: Ch40: 2Homework: Ch40: 12,13,16,22Due: April 17

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April 13 Particle in a well

40.3 Potential wellsA particle in a well of finite height (square-well potential):

Otherwise

0 0)(

U

LxxU

I II III

ExU

dx

d

m )(

2 2

22Schrödinger equation:

Bound states: When E<U0, the particle is more localized in the well.

1) Region II

.0for ,cossin)(

2 ,

2 222

2

LxkxBkxAx

mEkk

mE

dx

d

II

2) Region I and III

,

0 ,) Finite(

.or 0for ,)(2

,)(2 02

20

2

2

L xDe

xCe

LxxDeCeEUmEUm

dx

d

xIII

xI

xx

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Determining the constants in the equations by the boundary conditions and the normalization condition:

E

D

C

B

A

dxx

Lxdx

dψ

dx

d

Lxψ

xdx

dψ

dx

d

xψ

L xDe

L xkxBkxA ψ

xCe

IIIII

IIIII

III

III

xIII

II

xI

1)(

@

@

0@

0@

,

0 ,cossin

0 ,

2

Matching the functions at the boundary points is possible only for specific values of E, which are the possible energy levels of the system.

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Wave functions and energies of a particle in a well :• Outside the potential well, classical physics forbids the presence of the particle, while

quantum mechanics shows the wave function decays exponentially to approach zero.• The functions are smooth at the boundaries.• Each energy level for a finite well is lower than for an infinitely deep well of the same

width.

Applications: Nanotechnology: The design and application of devices having dimensions ranging from 1 to 100 nm. Using the idea of trapping particles in potential wells.Quantum dot: A small region that is grown in a silicon crystal, acting as a potential well.Storage of binary information.

Example 40.6.

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Read: Ch40: 3Homework: Ch40: 26,27,30Due: April 24

0 L

0

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April 20 Potential barriers and tunneling

40.4 Potential barriers and tunnelingPotential barrier: A place where the potentialenergy diagram has a maximum.

Square barrier:

U0 is the barrier height.

Otherwise 0

0 )( 0 LxU

xU

• Classically, if E < U0, the particle incident from the left is reflected by the barrier. Regions II and III are forbidden. In 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.

• The curve in the diagram represents a full solution to the Schrödinger equation. Movement of the particle to the far side of the barrier is called tunneling or barrier penetration.

• The probability of tunneling can be described by a transmission coefficient T.

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ExU

dx

d

m )(

2 2

22

,

0 ,

0 ,

)(2 ,

2 0

L xte

L xBeAe ψ

xree

EUmmEk

ikxIII

xxII

ikxikxI

2

2

@

@

0@

0@

rRr

tTt

B

A

Lxdx

dψ

dx

d

Lxψ

xdx

dψ

dx

d

xψ

IIIII

IIIII

III

III

• Transmission coefficient (T): The probability

for the particle to penetrate the barrier.• Reflection coefficient (R): The probability

for the particle to be reflected by the barrier.• T + R = 1

L

L

eU

E

U

E

EUE

LUT

2

00

1 If1

0

220 116

4

sinh1

Example 40.7Test 40.4

0 L

0

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Applications of tunneling:

Alpha decay:In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system.

Nuclear fusion:Protons can tunnel through the barrier caused by their mutual electrostatic repulsion.

Scanning tunneling microscope:

• The empty space between the tip and the sample surface forms the “barrier”.

• The STM allows highly detailed images of surfaces with resolutions comparable to the size of a single atom: 0.2 nm lateral, 0.001nm vertical.

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Read: Ch40: 3Homework: Ch40: 33,34,37Due: May 1

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April 24 Harmonic oscillator

40.5 The harmonic oscillator

The potential energy: 222

2

1'

2

1)( xmxkxU

The Schrödinger equation: ExU

dx

d

m )(

2 2

22

Exm

dx

d

m 22

2

22

2

1

2

Let us guess: 2

1 ,

2exp)( 2 E

mCCxBx

This is actually the ground state.

The actual solution:

,2 ,1 ,0 ,2

1

2exp

!2

/)( 2

nnE

xm

xm

Hn

mx

n

nnn

Hermite polynomials

22

,2 ,1 ,0 ,2

1

nnEn

• Ground state

• E

Example 40.8

Wave functions:

Energy levels:

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Probability density and comparison to Newtonian oscillators:

• The green curves represent probability densities for the first four states.• The blue curves represent the classical probability densities corresponding to the

same energies.• As n increases, the agreement between the classical and the quantum-mechanical

results improves.

Test 40.5

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Read: Ch40: 4Homework: Ch40: 38,40,41Due: May 1