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
- 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
Operators
- What is an operator in quantum mechanics?
- What is an expectation value in quantum mechanics?
- What is a commutator in quantum mechanics?
- 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
Hamiltonian
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)
All posts
- The minimum energy of a hydrogen atom in quantum mechanics
- 규격화된 파동함수의 상태는 시간의 변화에 무관하다
- Compton scattering
- The rest mass of a photon is zero.
- Electrons Cannot Be Constituents of the Nucleus
- Momentum operator in quantum mechanics
- Commutator of Momentum and Position
- Prove that the expectation value of momentum is always a real number.
- The Importance of the Relative Phase of the Wave Function
- Finding the Wave Function Eigenfunctions and Energy Eigenvalues in an Infinite Potential Well
- Properties of Commutators
- Commutation Relations of the Angular Momentum Operator
- What is an operator in physics (quantum mechanics)
- What is Dirac Notation?
- Hermitian Operator
- Proof that the expectation eigenvalue of a Hermitian operator is always real
- The Meaning of Eigenvalue Equations in Quantum Mechanics
- Two eigenvectors with different eigenvalues are orthogonal.
- Two operators with simultaneous eigenfunctions are commutative.
- For any arbitrary operator, always in the Hermitian form
- Ladder Operators for Angular Momentum
- Simultaneous Eigenfunctions of Angular Momentum
- The condition for the product of two Hermitian operators to be a Hermitian operator
- Relationship between simultaneous eigenfunctions of angular momentum and ladder operators
- There is no solution to the time-independent Schrödinger equation when the energy is less than the potential.
- What is Degeneracy of Wave Functions in Quantum Mechanics?
- Energy Levels in an Infinite Potential Well
- Solving Harmonic Oscillator Problems using the Operator Method: Definition of Ladder Operators
- Matrix Representation of Operators in Quantum Mechanics
- Matrix Representation of the Harmonic Oscillator Operator
- Matrix Representation of Angular Momentum Operator
- Probabilistic interpretation and normalization of the wave function in quantum mechanics
- Gram-Schmidt Orthogonalization Process in Quantum Mechanics
- Probability Flow
- Reflection and Transmission of Wave Functions
- Solution of the Schrödinger Equation for a Step Potential
- Solution of Schrödinger Equation for Finite Square Well Potential
- Solution of the Schrödinger Equation for a Potential Barrier
- Angular Momentum Operator in Spherical Coordinates
- Vectors and Inner Products in Quantum Mechanics
- Parity Operator
- Expectation Value in Quantum Mechanics
- de Broglie Equation and Matter Waves
- In Quantum Mechanics, what is a commutator?
- Derivation of the Schrödinger Equation
- Schrödinger Equation in Spherical Coordinates
- The eigenfunctions of the angular momentum operator are spherical harmonics.
- Ladder Operators of Angular Momentum in Spherical Coordinates
- Angular Momentum and Position/Momentum Commutation Relations
- Wave Function and Hilbert Space in Quantum Mechanics
- Zeeman Effect
- Proof of Heisenberg's Uncertainty Principle
- Angular Momentum Operator in Quantum Mechanics
- Position Operator in Quantum Mechanics
- What is the ladder operator in quantum mechanics?
- Hamiltonian Operator