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PHYS 30101 Quantum Mechanics
Lecture 7
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
Hermitian Operators
• They have real eigenvalues
• Eigenfunctions are orthonormal
• Eigenfunctions form a complete set
Summary of postulates
1. A quantum system has a wavefunction associated with it.
2. When a measurement is made, the result is one of the
eigenvalues of the operator associated with the
measurement.
3. As a result of the measurement the wavefunction
“collapses” into the corresponding eigenfunction.
4. The probability of a particular outcome equals the square
of the modulus of the overlap between the wavefunction
before and after the measurement.
Example of a “measurement”
polariser
50% transmitted
Photons of
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
Example of a “measurement”
Vertical
polarization
detector
Photons of
unpolarised light
Horizontal
polarization
detector
Birefringent crystal
(eg Icelandic spar)
Today:
1.4 Finish off with discussion on continuous eigenvalues
1.5(a) Commutators
1.5(b) Compatibility
If
then the physical observables they represent are said
to be compatible: the operators must have a common set of
eigenfunctions:
Example (1-D): momentum and kinetic energy operators have
common set of eigenfunctions
After a measurement of momentum we can exactly predict the
outcome of a measurement of kinetic energy.