PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

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PHY 102: Quantum Physics Topic 4 troduction to Quantum Theo

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PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory. Wave functions Significance of wave function Normalisation The time-independent Schrodinger Equation. Solutions of the T.I.S.E. The de Broglie Hypothesis. - PowerPoint PPT Presentation

Transcript of PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

Page 1: PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

PHY 102: Quantum Physics

Topic 4Introduction to Quantum Theory

Page 2: PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

•Wave functions

•Significance of wave function

•Normalisation

•The time-independent Schrodinger Equation.

•Solutions of the T.I.S.E

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The de Broglie Hypothesis

In 1924, de Broglie suggested that if waves of wavelength λ were associated with particles of momentum p=h/λ, then it should also work the other way round…….

A particle of mass m, moving with velocity v has momentum p given by:

h

mvp

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Kinetic Energy of particle

m

k

m

h

m

pKE

222

22

2

22

If the de Broglie hypothesis is correct, then a stream of classical particles should show evidence of wave-like characteristics……………………………………………

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Standing de Broglie waves

Eg electron in a “box” (infinite potential well)

V=0

V= V=

Electron “rattles” to and fro

V=0

V= V=

Standing wave formed

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Wavelengths of confined states

LkL

; 2

LkL

2 ;

Lk

L 3 ;

3

2

In general, k =nπ/L, n= number of antinodes in standing wave

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Energies of confined states

2

22222

22 mL

n

m

kE

12EnEn

2

22

1 2mLE

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Energies of confined states

12EnEn

2

22

1 2mLE

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)sin()sin(),( tkxAtxy

Particle in a box: wave functions

From Lecture 4, standing wave on a string has form:

Our particle in a box wave functions represent STATIONARY (time independent) states, so we write:

kxAx sin)(

A is a constant, to be determined……………

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Interpretation of the wave function

The wave function of a particle is related to the probability density for finding the particle in a given region of space:

Probability of finding particle between x and x + dx:

dxx2

)(

Probability of finding particle somewhere = 1, so we have the NORMALISATION CONDITION for the wave function:

1)(2

dxx

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Interpretation of the wave function

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Interpretation of the wave function

Normalisation condition allows unknown constants in the wave function to be determined. For our particle in a box we have WF:

L

xnAkxAx

sinsin)(

Since, in this case the particle is confined by INFINITE potential barriers, we know particle must be located between x=0 and x=L →Normalisation condition reduces to :

1)(0

2 L

dxx

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Particle in a box: normalisation of wave functions

1)(0

2 L

dxx 1sin0

22

L

dxL

xnA

L

xn

Lx

sin2

)(

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Some points to note…………..

So far we have only treated a very simple one-dimensional case of a particle in a completely confining potential.

In general, we should be able to determine wave functions for a particle in all three dimensions and for potential energies of any value

Requires the development of a more sophisticated “QUANTUM MECHANICS” based on the SCHRÖDINGER EQUATION…………………

Page 15: PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

The Schrödinger Equation in 1-dimension(time-independent)

)()()()(

2 2

22

xExxVdx

xd

m

KE TermPE Term

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Solving the Schrodinger equation allows us to calculate particle wave functions for a wide range of situations (See Y2 QM course)…….

Finite potential well

WF “leakage”, particle has finite probability of being found in barrier: CLASSICALLY FORBIDDEN

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Solving the Schrodinger equation allows us to calculate particle wave functions for a wide range of situations (See Y2 QM course)…….

Barrier Penetration (Tunnelling)

Quantum mechanics allows particles to travel through “brick walls”!!!!

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Solving the SE for particle in an infinite potential well

Lx0 0)( xV

So, for 0<x<L, the time independent SE reduces to:

)()(

2 2

22

xEdx

xd

m

0)(2)(

22

2

xmE

dx

xd

General Solution:

xmE

BxmE

Ax2/1

2

2/1

2

2cos

2sin)(

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xmE

BxmE

Ax2/1

2

2/1

2

2cos

2sin)(

Boundary condition: ψ(x) = 0 when x=0:→B=0

xmE

Ax2/1

2

2sin)(

Boundary condition: ψ(x) = 0 when x=L:

02

sin)0(2/1

2

LmE

A

2

222

2mL

nE

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L

xnAx

sin)(

In agreement with the “fitting waves in boxes” treatment earlier………………..

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Molecular Beam Epitaxy: Man-made potential wells for Quantum mechanical engineering

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Molecular Beam Epitaxy: Man-made potential wells for Quantum mechanical engineering

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Quantum Cascade Laser: Engineering with electron wavefunctions

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Scanning Tunnelling Microscope: Imaging atoms