PHYSICS 447

QUANTUM MECHANICS

Instructor: Purrington

Office: Stern 5050

email: danny@tulane.edu

TEXT: Griffiths, Introduction to Quantum Mechanics, 2nd ed.

TENTATIVE SYLLABUS/OUTLINE

Chapters 1-3, Griffiths []

Sections I-II:

I. Historical Overview--problems looming at the end of the 19th century; early 20th results

A. Blackbody radiation--classical theory, Planck (1900), Einstein (1905)

B. Specific heats

C. Photoelectric effect--Einstein (1905)

D. Atomic spectra--Bohr (1913); Zeeman effect, fine and hyperfine structure

E. Nuclear atom--Rutherford (1911)...(Geiger and Marsden)

F. Franck-Hertz (1914)

H. Stern-Gerlach experiment (1922)

[Pass out Chapter 1, Bransden and Jochain]

II. The Old Quantum Theory--Bohr, Sommerfeld, DeBroglie

A. Correspondence principle (Bohr, 1923)

B. Complementarity (Bohr, 1928)--wave-particle duality, uncertainty principle (Heisenberg, 1927; §1.6)

C. DeBroglie--wave properties of matter (1923-4), λ = h/p

Section III:

III. Wave mechanics (Schrodinger, 1926)

A. DeBroglie, wave-particle duality, Taylor (1906)--double slit exp. for photons at very low illumination ( Bransden§2.1).

B. Schrodinger equation (1926) for the wave function ψ:

(-ℏ^{2}/2m)∂^{2}ψ(x)/∂x^{2} + Vψ(x) = iℏ∂ψ/∂t

1, Interpretation of ψ, probablistic interpretation (Born, 1926); |ψ| ^{2}. Concept of a state. The wave function or state function as giving all of the information that can be known about the system.

|ψ| ^{2} dx as prob. of finding a particle in dx

coordinate space (position rep.) vs. momentum space (momentum rep.)

Normalization ∫|ψ| ^{2} dx=1 reflecting the probability of finding the particle somewhere. Square integrability.

2. Measurement process; collapse of wave function (Bohr). Copenhagen interpretation vs. time development from SE.

3. Superposition (re collapse); quantum ontology; Schrodinger’s Cat--Young’s experiment; linear DE. Superposition as mathematical property of Hilbert space.

4. Schrodinger equation as second order in x, first order in t. Why? For a plane wave, the time-dependent SE leads to E=p^{2}/2m. If second-order in time, leads to E^{2} ∝ p^{2}. Which is the correct relativistic result but yields neg. prob. density. Note
that ψ will be complex.

5. Conservation of normalization or probability density. Definition of prob. current density (continuity equation). See Merzbacher, Chap. 4:

Briefly, if we define ρ = |ψ|^{2} then conservation of probability density (analogous to conservation of charge) requires that ρ satisfy a continuity equation:

∂ρ/∂t + ∇j =0

This will be shown to require that j =-iℏ/2m [∇ψ* ψ -(∇ψ)*ψ]

Section IV, A-E:

IV. Some linear algebra. The “state” as “state vector,” that is, as a vector in a complex vector space (inner product space) called a “Hilbert space.” The operators on the Hilbert space represent linear transformations and observables.

A. Definition of a vector space. Linear operators on a Hilbert space of square-integrable functions. These can be ψ(x,t), ψ(x), or, for example, a function of momentum, φ (p).

B. Expectation values of physical quantities. Meaning of an expectation value as the average of a large number of measurements on identically prepared systems.

<x> = ∫x |ψ| ^{2} dx = ∫ψ* x ψ dx

So we have vectors or states and operators on them. The vector space has, among other things, the property of closure: Oψ=ψ’, where ψ’ is also in the vector space.

Some of the operators will correspond to observables. They will turn out to be Hermitian operators.

C. The Schrodinger equation as an operator equation

Hψ=iℏ∂ψ /∂t

where H is an operator related to the classical Hamiltonian, H = p^{2}/2m + V, which is the energy E, if V is not time dependent and not velocity dependent. Here, however, H is an operator, as will be seen when we find the form for the momentum operator p.

D.The momentum operator (§1.5): p--> -iℏ ∂ /∂x

X,P,H, etc., are operators on a Hilbert space of functions which are the states of the system. Thus we need more linear algebra, because while the expectation value of the operator O may be written <O> = ∫ψ*O ψ dx , in a “position representation”. But this is a special case of a more general result, which we can write as an inner product:

<O>=(ψ, Oψ)

which, itself, is a special case of (φ_{1}, φ_{2}). The inner product is the way in which two vectors are combined to yield a scalar. It is, of course, analogous to a.b in a Euclidean space.

E. Time-independent Schrodinger equation (§2.1)

1. We separate ψ(x,t) as follows: ψ(x,t)=ψ(x)φ(t), and insert it into the T-D Schrodinger equation. The result is :

-ℏ2/2m ∂^{2}ψ/∂x^{2} + Vψ(x) = Eψ(x),

or, in terms of p=-iℏ∂/∂x,

(p2/2m + V)ψ(x)=Eψ(x)

Such states ψ(x) are “stationary states” because their time dependence is just e^{-iωt}, which is just a phase factor (modulus 1). It doesn’t affect |ψ|^{2} = ψ*ψ.

We can write the previous equation as Hψ=Eψ, which is an “eigenvalue equation.”

Section IV, F-L:

F. Eigenvalues and eigenfunctions

The time-independent Schrodinger equation in the form Hψ=Eψ, is an eigenvalue equation. ψ(x) is the eigenfunction and E is the eigenvalue. In general, an eigenvalue equation is of the form Oφ=λφ, where λ is a scalar constant. In general, solutions exist only for specific values of λ. These specific values are the spectrum of the operator O, which may be discrete or continuous.

Later we will show that eigenvalues of Hermitian operators are real, which is a necessary condition that something be observable or measureable (dyanimcal variables). Non- Hermitian operators will play a role in the theory as well (e.g., unitary operators), representing linear transformations.

The result of the measurement of a dynamical variable will always be one of the eigenvalues of the associated Hermitian operator.

H. Uncertainty principle (Heisenberg, 1927)

1. Proof of the uncertainty principle, §3.5

(Δ x)^{2} = <(x - <x>)^{2}> = <x^{2}> -<x>^{2}

J. Hermitian operators, measurement, observables (§3.2)

K. Complete, orthonormal sets of basis vectors.

1. Transformation from one basis to another. Unitary transformations. Coordinate and momentum representations. Expanding a state in terms of a complete set of orthonormal basis f unctions

2. Abstract formalism, Dirac notation (§3.6); matrix mechanics. Discreet and continuous spectra of Hermitian operators (§3.3) Distributions, Dirac δ-functions

Suppose that the position of a particle is specified precisely. What is the form of its wavefunction, which localizes it at the point x = x_{o}? It must satisfy ∫|ψ|^{2} dx = 1, and yet the probability of finding the particle anywhere except x = x_{o} must be zero. What kind of function has that property? No ordinary function. We call it, however, the Dirac δ- function, and its justification is found in the theory of distributions. It must have the property that:

∫ δ (x-x_{o}) dx =1 (the area under the curve y=δ(x) is unity)

as long as x_{o} is in the interval of integration.

Since it is peaked and is infinite at x=xo (in order to have a finite integral in spite of being defined only at a point), if we multiply some function f(x) by δ (x-xo) and integrate over x, we must get the following result:

∫ f(x) δ(x-x_{o}) dx = f (x_{o})

3. Wave packets; minimum uncertain wave packet

L. One-dimensional examples (Chapter 2)

1. Free particle, wave packets

2. Infinite square well, particle in a finite well; barrier penetration, reflection

3. Linear harmonic oscillator (§2.3).

a. Ladder operators, commutators, canonical commutation relations.

b. Direct solution of the Schrodinger equation.

Chapter 4, Griffiths []

V . Quantum Mechanics in Three Dimensions (Chapter 4); product states

A. Infinite spherical well

B. Hydrogen atom; angular momentum and spin; rotation group

Chapter 5, Griffiths []

VI. Multiparticle systems (atoms, etc.) (Chapter 5)

A. Identical particles; quantum statistics

B. Atoms, solids (brief)

Chapter 6, Griffiths []

VII. Time-independent Perturbation Theory (Chapter 6)

A. Zeeman effect

B. Stark effect

Chapters 7-12, Griffiths [];

VIII. Chapters 7-10; some highlights as time permits

A. Time-dependent perturbation theory

B. Variational principles

C. WKB approximation

D. Adiabatic approximation

IX. Scattering (Chapter 11)

X. Foundations and Interpretation. What does it mean? (Chapter 12)

A. EPR; Bell’s Theorem

B. Many worlds interpretation

Sept. 24 Last day to drop without record

Oct. 26 Last day to drop

Mid-term Exam, Chapters 1-3, around Oct. 15.

Final Exam