PHYS 623 Introduction to Quantum Mechanics II
Spring 2016, University of Maryland
Prof. Ted Jacobson

Notes

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Thursday, Feb. 25

$1\otimes \mathrm{1/2 }$$\mathrm{<mo>\otimes \; </mo>1/2}=\mathrm{}$$1\otimes 1=2\oplus 1\oplus 0$) = $1\otimes 1=2\oplus 1\oplus 0$ = $1\otimes 1=2\oplus 1\oplus 0$$1\otimes 1=2\oplus 1\oplus 0$ : (dimension check: 3 x 2 x 2 = 5 + 3 + 3 +1). Contains one singlet, two triplets, and a quintuplet
$\mathrm{<br>\; <br>}\mathrm{1/2 }$$\mathrm{<mo>\otimes \; </mo>1/2}$$1\otimes 1=2\oplus 1\oplus 0$) $1\otimes 1=2\oplus 1\oplus 0$) = $\mathrm{(1}\otimes \mathrm{}$1) ⊕1 ⊕1 ⊕0 = $1\otimes 1=2\oplus 1\opl$$1\otimes 1=2\oplus 1\oplus 0$ $1\otimes 1=2\oplus 1\oplus 0$$\mathrm{}$: (dimension check: 2 x 2 x 2 x 2 = 5 + 3 + 3 + 3 + 1 + 1). Contains two singlets, among other things. Singlets are invariant under rotation so can be more protected from disturbance by external fields. The two states of a protected qbit could thus be these two singlets.

Hyperfine structure of Cs-133 defines the SI unit of one second: 9, 192, 631, 770 periods of the radiation from this transition. The spin of this nucleus is 7/2, and the atom is an alkali metal, so has J = S = 1/2. There are thus two hyperfine levels, corresponding to F = 4 and F = 3.

Selection rules for tensor operators and the Wigner Eckart theorem.

Solved Spring '16 qualifier symmetry problem

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Tuesday, Feb. 23

Vector operators, Projection Theorem, Tensor operators

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Thursday, Feb. 18

1. Stark effect, both quadratic and linear.

2. Zeeman effect, including description of his original experiment.

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Tuesday, Feb. 16

1. Brief review of addition of angular momentum. "Irreducible representation" means a representation with no invariant proper subspace. For example, in the homework this week, you consider the product of two spin-1 representations, each 3 dimensional, which decomposes into the sum of spin-2, spin-1, and spin-0 representations, $1\otimes 1=2\oplus 1\oplus 0$. When acted on by rotations, each of the latter transform into themselves. For example the spin-2 subspace rotates into itself, so it is an invariant, 5 dimensional subspace of the $1\otimes 1$ product representation.

2. QED (quantum electrodynamics) origin of the Lamb shift: it comes from vacuum polarization and self-energy corrections. The existence of vacuum polarization means that the charge of the nucleus is partially screened, so when an electron is closer to the nucleus it feels a stronger charge, lowering the energy. This effect is strongest for s-states, but also felt in other orbitals. It is less important however than the self-energy correction, which comes from the interaction of an electron with the electromagnetic field that it sources. This leads to an extra "jiggling" of the electron, whose effect is similar to that of the Darwin term from the Dirac equation (see text), but smaller by a factor$\alpha\, \ln \alpha$$\alpha\,\ln\,\alpha$. In effect,  the electron feels the local potential averaged over a small spherical region. The average of the potential over a small sphere is the value at the center, plus a correction whose leading order contribution is proportional to the Laplacian of the potential. For the Coulomb potential this vanishes except at the origin, so this effect is only important for s-states. It reduces the effective depth of the potential, raising the energy level. 

3. Hyperfine structure. Emphasized origin of the $1/M_N$ dependence of the nuclear magnetic moment: a rotating classical body whose mass density and charge density have a constant ratio  throughout the body has ratio of magnetic moment to angular momentum given by $\mu/L= q/mc$, where $q$ and $m$are the charge and mass. (The factor $1/c$ is present when using Gaussian units.) In view of this, the magnetic moment of a body like an electron or a nucleus is written as $\mu = g q I/2mc$, where $I$ is the spin (intrinsic angular momentum) of the body,  and the $g$ factor" is determined by the nature and structure of the body. Note that a nuclear magnetic moment is suppressed compared to the electron magnetic moment by the small ratio of the electron mass to the nuclear mass.

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Thursday, Feb. 11

1. Time reversal operation on spinors relies on the identity $\sigma_i \sigma_2 = -\sigma_2 \sigma_i^*$. Viz, suppose $\chi$ is a spin-up state in the $n^i$ direction, so that $n^i\sigma_i \chi = \chi$. Then $n^i \sigma_i (\Theta \chi) = n^i \sigma_i (-i\sigma_2 \chi^*) = i \sigma_2 (n^i \sigma_i \chi)^* = i \sigma_2\chi^* = -\Theta\chi$, so $\Theta\chi$ has the opposite spin from that of $\chi$.

2. The velocity operator for a Dirac particle is $\dot x^i=\alpha^i$. A free particle energy eigenstate with energy $mc^2$ satisfies $\langle \alpha^i\rangle=0$, as expected. But there is a surprise: $\alpha^i\alpha^i = 3I$ as an operator, so $\langle \dot x^i\dot x^i\rangle = 3$. The rms velocity is thus $\sqrt{3}$ times the speed of light! Note, however, that the different components of velocity don't commute with each other.

3. Relativistic corrections to the electron spectrum in the Coulomb potential.

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Tuesday, Feb. 9

1. Discussion of 1st and 2nd order perturbation theory.

2. Dirac equation.

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Thursday, Feb. 4

1. Perturbation theory for stationary states.

2. Review of ladder operators for the harmonic oscillator.

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Tuesday, Feb. 2

1. How to think about commutators of symmetry generators with other operators.

2. Translation, discrete translation, and time reversal symmetries.

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Thursday, Jan. 28

1. Dirac notation review

2. Symmetries: space inversion a.k.a. parity.

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Tuesday,Jan. 26

SNOW DAY

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