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

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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

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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

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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

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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

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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)

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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)

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Lecture 1

Summary

• Foundations & postulates

• Schrodinger equation

• Scattering in 1D

Next Time

• Bound states

• WKB approximation

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