Quantum Mechanics

To properly understand and study quantum mechanics, one must absolutely study linear algebra. It would be ideal to study Fourier analysis and Hilbert spaces as well, but if one had to choose just one most important subject among them, it would be linear algebra. I strongly recommend taking the linear algebra courses offered by the mathematics department.

Quantum Effects

• The rest mass of a photon is $0$
• Compton scattering
• Minimum energy of a hydrogen atom
• An electron cannot be a component of the nucleus
• De Broglie’s relation and matter waves
• Zeeman effect

Theoretical Framework of Quantum Mechanics

• Vectors and inner product in quantum mechanics
• Dirac notation
• Time-independent/dependent Schrödinger equation derivation
  • Schrodinger Equation in Spherical Coordinates
• Meaning of eigenvalue problem in quantum mechanics
• Proof of the uncertainty principle

Wave Function

• Wave function and Hilbery space
• Probabilistic interpretation and normalization of wave function
• Probability current
• What is the degeneracy of a wave function?
• Gram-schmidt orthogonalization procedure
• There is no solution to the time-independent Schrödinger equation when the energy is less than the potential
• A normalized wave function is independent of time changes

Time-independent Schrödinger Equation

1D Potential

• Step function potential
• Finite well potential
• Barrier potential
• Delta function potential

Operators

• What is an operator in quantum mechanics?
  • Position operator $X$
  • Parity operator
  • Ladder operator $L_{\pm}$
• What is an expectation value in quantum mechanics?
• What is a commutator in quantum mechanics?
  • Properties of commutators
• Two operators with simultaneous eigenfunctions are commutable
• Matrix representation of Operator

Hermitian Operators

• Hermitian operator
• The expectation value/eigenvalue of a Hermitian operator is always real
• Different eigenfunctions of a Hermitian operator are orthogonal

Momentum

• Momentum operator
• Commutator of position and momentum $[x, p] = i\hbar$
• The expectation value of momentum is always real

Angular Momentum

• Angular momentum operator
  • Commutation relations
  • Commutation relations of angular momentum with position/momentum
  • Angular momentum operator in spherical coordinates
  • Matrix representation
• Ladder operator $L_{\pm}$
  • Simultaneous eigen functions $\ket{\ell, m}$
  • The simultaneous eigenfunctions of the angular momentum operator are spherical harmonics
  • Relationship between eigenfunctions and ladder operators

Hamiltonian

• Hamiltonian operator $H$

References

• Stephen Gasiorowicz, 양자물리학(Quantum Physics, 서강대학교 물리학과 공역) (3rd Edition, 2005)
• David J. Griffiths, 양자역학(Introduction to Quantum Mechanics, 권영준 역) (2nd Edition, 2006)