aqp_lecture1
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Advanced Quantum Physics
Nigel Cooper
http://www.tcm.phy.cam.ac.uk/~nrc25/aqp/
Lecture 1
Today
• Introduction to the course
• Foundations & postulates
• Schrodinger equation
1
Lecture 1
Aim of the course
Building upon the foundations of wave mechanics,this course will introduce and develop the broad fieldof quantum physics including:
• Quantum mechanics of point particles
• Approximation methods
• Foundations of atomic, molecular, and solid statephysics
• Light-matter interactions
• Basic elements of quantum field theory
2
Lecture 1
Prerequisites
This course will assume familiarity with NST IBQuantum Physics (or equivalent):
• Failure of classical physics
• Wave-particle duality, and the uncertainty principle
• The Schrodinger equation+ solutions (barriers, wells, SHO, Hydrogen)
• Dirac notation
• Operator methods
• Angular momentum and spin
• Indistinguishable particles
3
Lecture 1
Practicalities
• 24 Lectures, M.W.Th. 9
• Two handouts, and two problem sets.
• The handouts include more material than thesyllabus. (For the most part, non-examinablematerial will be marked as “Info”.)
• All materials will be available from:
www.tcm.phy.cam.ac.uk/~nrc25/aqp.html
Please inform me of any typos/errors.
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Lecture 1
Books
• Quantum Physics, S. Gasiorowicz
• The Physics of Atoms and Quanta,
H. Haken and H. C. Wolf
• Quantum Mechanics: A New Introduction,
K. Konishi and G. Paffuti
• Quantum Mechanics: Non-Relativistic Theory,
Volume 3, L. D. Landau and L. M. Lifshitz
• Quantum Mechanics, F. Schwabl
• Principles of Quantum Mechanics, R. Shankar
• Problems in Quantum Mechanics, G. L. Squires
5
Lecture 1
Today
• Introduction to the course
• Foundations & postulates
• Schrodinger equation
6
Lecture 1
Foundations of QM
Historically, origins of quantum mechanics can betraced to failures of 19th Century classical physics:
• Black-body radiation
• Photoelectric effect
• Compton scattering
• Atomic spectra: Bohr model
• Electron diffraction: de Broglie hypothesis
7
Lecture 1
Schrodinger’s equation
Plane wave: Ψ(x, t) = Aei(kx−ωt)
Planck: E = ~ω
de Broglie: p = h
λ= hk
2π = ~k
E =p2
2m⇒ ~ω =
~2k2
2m
i~∂tΨ(x, t) = − ~2
2m∂2xΨ(x, t)
+ Potential, 3D E =p2
2m+ V (r, t)
i~∂tΨ(r, t) = −~2
2m∇2Ψ(r, t) + V (r, t)Ψ(r, t)
8
Lecture 1
Postulates of quantum mechanics
1. The state of a quantum mechanical system iscompletely specified by the complex wavefunctionΨ(r, t).
Ψ∗(r, t)Ψ(r, t) dr is the probability that a particlelies in volume element dr ≡ ddr at time t.
Normalization:
∫
|Ψ(r, t)|2dr = 1.
2. To every observable in classical mechanics therecorresponds a linear Hermitian operator, A.
The result of a measurement of A is a number a,where a is one of the eigenvalues AΨ = aΨ.
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Lecture 1
3. If system is in a state described by Ψ, andobservable A is measured, the probability ofobtaining the value ai (where AΨi = aiΨi) is
P (ai) =
∣
∣
∣
∣
∫
∞
−∞
Ψ∗
i(r)Ψ(r)dr
∣
∣
∣
∣
2
= |〈Ψi|Ψ〉|2
A measurement of Ψ that leads to eigenvalue aicauses the wavefunction to “collapse” to thecorresponding eigenstate Ψi.
4. Between measurements, the wavefunction evolvesaccording to the time-dependent Schrodingerequation,
i~∂tΨ = HΨ
.
H = −~2
2m∇
2 + V (r, t)
10
Lecture 1
Summary
• Foundations & postulates
• Schrodinger equation
• Scattering in 1D
Next Time
• Bound states
• WKB approximation
19
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