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PHYS 30101 Quantum Mechanics
Lecture 10
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: www.man.ac.uk/dalton/phys30101
Plan of action
1. Basics of QM
Will be covered in the following order:
2. 1D QM
1.1 Some light revision and reminders. Infinite well
1.2 TISE applied to finite wells
1.3 TISE applied to barriers – tunnelling phenomena
1.4 Postulates of QM
(i) What Ψ represents
(ii) Hermitian operators for dynamical variables
(iii) Operators for position, momentum, ang. Mom.
(iv) Result of measurement
1.5 Commutators, compatibility, uncertainty principle
1.6 Time-dependence of Ψ
1.7 Degeneracy
1.7 Degeneracy
Orthonormality: it is always possible to construct a set of
orthonormal eigenfunctions from a set of non-orthonormal
eigenfunctions (“Schmidt orthogonalization” – see Rae)
Compatibility: If
then a common set of
eigenfunctions exist.
But we will show an eigenfunction of Q is not necessarily an
eigenfunction of R.
Nevertheless a set of eigenfunctions can be found that are
common to both operators
Syllabus
1. Basics of quantum mechanics (QM)
Postulate, operators,
eigenvalues & eigenfunctions, orthogonality & completeness, time-dependent
Schrödinger equation, probabilistic interpretation, compatibility of
observables, the uncertainty principle.
2. 1-D QM Bound states, potential barriers, tunnelling phenomena.
3. Orbital angular momentum
Commutation relations, eigenvalues
of Lz and L2, explicit forms of Lz and L2 in spherical polar coordinates, spherical
harmonics Yl,m.
4. Spin
Noncommutativity of spin operators, ladder operators, Dirac notation,
Pauli spin matrices, the Stern-Gerlach experiment.
5. Addition of angular momentum
Total angular momentum
operators, eigenvalues and eigenfunctions of Jz and J2.
6. The hydrogen atom revisited
Spin-orbit coupling, fine structure,
Zeeman effect.
7. Perturbation theory
First-order perturbation theory for energy levels.
8. Conceptual problems
The EPR paradox, Bell’s inequalities.
The main points so far:
There is an (hermitian) operator for every dynamical variable
(position, momentum, energy…). Each operator has a complete set
of orthonormal eigenfunctions with real eigenvalues:
In order to predict the result of a measurement of “q” on a
general wavefunction Ψ we need to describe Ψ as a linear
combination of the eigenfunctions of the corresponding
operator:
Ψ = a1φ1 + a2φ2 + a3φ3 ….. + anφn …
where
The measurement forces Ψ to decide which eigenfunction it
is going to “collapse” into (think about unpolarised photons
and a polariser). It will collapse into the state φm with
probability |am|2 and return the value “qm”.
Expectation value (mean value) of q:
Example of a “measurement”
polariser
50% transmitted
Unpolarised light
100% polarised
Describe each photon as a linear
combination of eigenfunctions of dynamic
variable being measured:
After measurement
photon collapses into
the corresponding
eigenfunction
= 50% VERTICAL + 50% HORIZONTAL
After measurement the photon has no memory of its polarization
state before the polariser.
All subsequent Vertical/Horizontal measurements of transmitted
photon will give the definite result: Vertical
If
then there exists a common set of eigenfunctions:
A measurement of “q” will collapse Ψ into one of the eigenfunctions φm
and return the result “qm”
A subsequent measurement of “r” will now give the definite result “rm”
Any further measurement of “q” or “r” for this wavefunction is
exactly predictable – the answers will always be “qm” and “rm”.
In this case ΔqΔr = 0
Degeneracy
If
then there exists a common set of eigenfunctions
but if there is some degeneracy:
and a measurement returns the value “q” the wavefunction collapses
into an unknown combination of eigenfunctions that have this
eigenvalue:
A subsequent measurement of “r” will then force the intermediate
wavefunction to collapse into one of the common eigenfunctions,
leaving “q” unchanged and returning the eigenvalue “rn” (say).
Any further measurement of “q” or “r” for this wavefunction is
exactly predictable – the answers will always be “q” and “rn”.
In this case ΔqΔr = 0
Example: n=2 level of H-atom
L=1
L=0
n=2
(2p level)
(2s level)
m=0
Measure L2
m=+1
m=0
m=-1
(has eigenvalues L(L+1) ħ2 )
If result is “2ħ2” then we know L=1 but don’t know what “m” will
be before the measurement of Lz (its eigenvalues are “m”)
Measure Lz, and we force atom to decide its orientation “m”
(ie the value of the projection of L on our chosen z-axiz)
All subsequent measurements of L2 and Lz will give same results.
3. Orbital Angular Momentum
One of the most important aspects of QM with implications in
particle, nuclear, atomic, molecular, laser physics.
Angular momentum is always quantised because of periodic
boundary conditions such as Ψ(r, θ, φ) = Ψ(r,θ, φ+2π).
[Background (beyond this course): Paul Dirac (in 1928) “guessed” the
relativistic version of the Schrödinger Equation following principle of
postulate 3. This is called the Dirac Equation.
•“Intrinsic” spin ½ particles emerged from his theory in a natural way
We deal with intrinsic spin in this course
•Intrinsic magnetic moment correctly predicted
•Proton and neutron were not “Dirac” particles but have sub-structure
•Antiparticles predicted. Positron discovered in 1933 (Anderson)]
Useful formulae
TDSE – time dependent
Schrödinger Equation
Vector operators in spherical polar
coordinates
Angular momentum
operators in spherical polars
TISE – time independent S.E.