_"What is the main dynamical law in quantum mechanics?"_ A very short answer is the Schrödinger equation.
The Schrödinger equation dictates how the quantum state of a physical system changes with time. The quantum state is represented by $\Psi$. There are two "versions" of the Schrödinger equation (SE); the time dependent SE and the time independent SE.
The time dependent SE is given by:
$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r}, t) = \hat{H}\Psi(\mathbf{r}, t)$
The time independent SE is given by:
$\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r})
$
The time independent SE is when the potential energy operator that is part of the Hamiltonian is independent of time. The "$E
quot; represents the allowed energies of the system. My professor called this the "eigenvalue problem of the Hamiltonian". What happens is that when the Hamiltonian acts on $\psi(\mathbf{r})$, it spits out the eigenvalues which represent the allowed energies of the system.
> I need to understand eigenvalues better. I know the right place: https://youtu.be/PFDu9oVAE-g?si=jsZSzYywPTte0Nfs
Physically meaningful wavefunctions for a given Hamiltonian are either stationary states (that is, they have a specific energy and evolve over time in a specific way) or they can be expressed as a sum of stationary states
**Linear algebra digression:** homogenous systems are guaranteed to have a trivial solution. For a homogenous system, you have a non trivial solution only if $\det(A) = 0$.
### How to spot a Hermitian (conjugate transpose, hermitian conjugate, adjoint)
- Diagonal elements real
- Elements $ij$ and $ji$ are complex conjugates of each other
### Unitary matrices
- These describe moving between coordinate systems
- $U^\dagger AU$ - describes how A looks if I write it in the basis of vectors that are the columns of U
## Particle in a box (particle in infinite potential well)
- Prototype confined system in 1D
- Goal is to find the allowed energies and stationary states (eigenvectors of the Hamiltonian) for particle of mass m confined to a region $x \in [0, L]$ by infinitely high potential barriers
- $V(x) = 0$ for $x \in [0, L]$ and $\infty$ otherwise (piecewise constant)
- Solving the time independent Schrödinger wave equation for the above mentioned conditions for $V(x)$ we get the following form: $\phi(x) = Ae^{ikx} + Be^{-ikx}$ (this is a plane wave)
- $k = \sqrt{\frac{2 m E}{\hbar ^2}}$
- _"The wavefunction is the superposition of a wave going forward and a wave going backward"_
- Quantised quantities:
- $n = \frac{kL}{\pi}$ (you arrive at this by applying the boundary condition to the wave function)
- $E_n = \frac{n^2 \hbar ^2 pi^2}{2 m L^2}$ ($\hbar$ is reduced Planck constant)
- The further up you go, the further apart the energy states get
- You get sharp energy levels; confinement is quite useful