# Physics GRE 5. Quantum Mechanics and Atomic Physics Essay

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5.1 Average Value of Operator, “A”, on state psi

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5.2 Position-Probability Normalization of state psi

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5.3 Property of Hermitian Operator “Q” acting on states f and g

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5.4 Position and Momentum Operators

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5.5 Probability of Obtaining nth eigenvalue

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5.6 Expectation (Average) Value

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5.7 Inner product of and a and b (Bracket Notation)

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5.8 Bracket Notation Inner product commutative property

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5.9 Hermitian Conjugate Condition for inner product

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5.10 f(x), inner product relation for continuum states

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5.11 inner product relation for states f,g with continuous position locations x

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5.12 Schrodinger Equation

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5.13 Hamiltonian Operator

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5.14 Schrodinger, Eigenvalue Equation

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5.15 Position, Momentum commutation relation

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5.16 Uncertainty relation of operators A and B based off of their commutator

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5.17 Statistical variance/ Uncertainty

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5.18 Uncertainty Product for position and momentum

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5.19 Order of Magnitude estimate of position and momentum uncertainty

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5.20 Energy and Time order of magnitude momentum and position uncertainty

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5.21 One Dimensional Harmonic Oscillator Hamiltonian

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5.22 One dimensional Harmonic Oscillator Hamiltonian using raising and lowering operators

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5.23 Commutation relation of lowering and raising operators

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5.24 One Dimensional Harmonic Oscillator Energy Eigenvalues

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5.25 Virial’s Theorem for Harmonic Oscillator

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5.26 Three Dimensional Harmonic Oscillator Energy Eigenvalues

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5.27 Eigenfunctions and Energy Eigenvalues of a free particle

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5.28 De Broglie Formula

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5.29 Einstein’s relation (energy of a free particle)

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5.30 General Hamiltonian in Three Dimensions

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5.31 3 Dimensional, position-momentum commutation along same direction

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5.32 3 Dimensional, position-momentum commutation along different directions

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5.33 Radial Wave function Normalization conditions

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5.34 x component of orbital angular momentum

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5.35 y component of orbital angular momentum

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5.36 z component of orbital angular momentum

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5.37 Commutator relation of orthogonal components of orbital angular momentum

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5.38 Total orbital angular momentum

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5.39 Z component of angular momentum acting on a spherical harmonic state

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5.40 Total orbital angular momentum acting on a spherical harmonic state

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5.41 Allowed values of m (magnetic moment) in terms of l (orbital angular momentum)

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5.42 Orthonormaility of Spherical Harmonics

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5.43 Hamiltonian of Hydrogen Atom

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5.45 Un-normalized Ground state of Hydrogen Atom

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5.46 Hydrogen Ground State Energy

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5.47 nth Energy Eigenvalues of Hydrogen considering quantum number n only

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5.48 fine structure constant

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5.49 spin up and down along x in the z basis

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5.50 spin up and down along y in z basis

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5.51 Raising and Lowering Operators for spin 1/2 in terms of spin along x and y

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5.54 Adding spins s and s’

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5.55 Adding z component of spin quantum number m

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5.56 Singlet State a system of 2 spin 1/2 particles

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5.57 Perturbation Theory, 1st order energy Shift

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5.58 Perturbation Theory, 2nd order energy shift

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5.59 Eigenvalues of Spin-Orbit Operator, fine structure perturbation

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5.60 Change in Hamiltonian from Stark Effect