Quantum Mechanics II

1

Schrodinger Equation and The Wave Function

In this lesson you will learn about the time-dependent and independent Schrödinger equation (1d, 3d), how it is derived and what you can do with it.

Related formulas

Time independent Schrödinger equation (3d)

$$ W \, \mathit{\Psi} ~=~ - \frac{\hbar^2}{2\class{brown}{m}} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi} $$

Time-dependent Schrödinger equation (3d)

$$ i \, \hbar \, \frac{\partial \mathit{\Psi}}{\partial t} ~=~ - \frac{\hbar^2}{2\class{brown}{m}} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi} $$

2

Bra-Ket Notation: All You Need to Know

Here you will learn how Bra-Ket notation (Dirac notation) is defined, which computational rules exist for it and which advantages this notation brings.

3

Hermitian Operators (Matrices) in Quantum Mechanics

Learn what Hermitian operators and matrices are and what three important properties they have. Examples are also made.

4

Angular Momentum in Quantum Mechanics: Commutators and Eigenvalues

Here you will learn about angular momentum in quantum mechanics, its commutators, and how angular momentum states and eigenvalues are generated using ladder operators.

Related formulas

Angular momentum commutator (Lx and Ly)

$$ [ L_{\text x}, L_{\text y} ] ~=~ \mathrm{i} \, \hbar \, L_{\text z} $$

Commutator (Angular and Linear Momentum)

$$ [ \hat{L}_{\class{red}{i}}, \, \hat{p}_{\class{green}{j}} ] ~=~ \varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}} \, \mathrm{i} \, \hbar \, \hat{p}_{\class{blue}{k}} $$

Commutator for All Angular Momentum Component Operators

$$ [ L_{\class{red}{i}}, \, L_{\class{green}{j}} ] ~=~ \varepsilon_{\class{red}{i}\class{green}{j}\class{blue}{k}} \, \mathrm{i} \, \hbar \, L_{\class{blue}{k}} $$