PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory

Author
adamcarpenter 
Category
Documents

view
44 
download
1
Embed Size (px)
description
Transcript of PHY 102: Quantum Physics Topic 4 Introduction to Quantum Theory
PHY 102: Quantum Physics
Topic 4Introduction to Quantum Theory
•Wave functions
•Significance of wave function
•Normalisation
•The timeindependent Schrodinger Equation.
•Solutions of the T.I.S.E
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
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 wavelike characteristics……………………………………………
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
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
Energies of confined states
2
22222
22 mL
n
m
kE
12EnEn
2
22
1 2mLE
Energies of confined states
12EnEn
2
22
1 2mLE
)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……………
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
Interpretation of the wave function
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
Particle in a box: normalisation of wave functions
1)(0
2 L
dxx 1sin0
22
L
dxL
xnA
L
xn
Lx
sin2
)(
Some points to note…………..
So far we have only treated a very simple onedimensional 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…………………
The Schrödinger Equation in 1dimension(timeindependent)
)()()()(
2 2
22
xExxVdx
xd
m
KE TermPE Term
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
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”!!!!
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)(
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
L
xnAx
sin)(
In agreement with the “fitting waves in boxes” treatment earlier………………..
Molecular Beam Epitaxy: Manmade potential wells for Quantum mechanical engineering
Molecular Beam Epitaxy: Manmade potential wells for Quantum mechanical engineering
Quantum Cascade Laser: Engineering with electron wavefunctions
Scanning Tunnelling Microscope: Imaging atoms