# 0x101 Quantum Mechanics

## Foundation

### Experiments

The Stern-Gerlach experiment SG experiment measures the z-component of the magnetic momentum $$\mu$$ of random oriented silver atoms, which are expected to distribute continuously between $$[ -|\mu|, |\mu| ]$$, but are actually realized in the two quantized spots.

Furthermore, concating multiple SG apparatus show that determining x,z component simutaneously is impossible. Measuring one will destroy info of the other.

### Schrödinger Equation

The wave function $$\Psi{(x,y)}$$ encodes all information about the state of a particle. Wave function can be added, multiplied by a complex number and taken inner product, therefore they are in the Hilbert space

The wave function can be obtained by solving the Schrödinger's equation

$i \hbar \frac{\partial \Psi}{\partial t} = - \frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V \Psi$

The wave function should satisfy the normalization constraint, which states

$\int_{-\infty}^{\infty} |\Psi(x,t)|^2 dx = 1$

The wave function has a property that it automatically preserves the normalization

$\frac{d}{dt} \int |\Psi(x,t)|^2 dx = 0$

therefore we only need to do the normalization at $$t=0$$

The expected position of a particle in state $$\Psi$$ is

$\langle x \rangle = \int x |\Psi(x,t)|^2 dx = \int \Psi^* [x] \Psi dx$

The expected momentum is

$\langle p \rangle = \int \Psi^* [-i \hbar (\frac{\partial}{\partial x})] \Psi dx$

where $$[x], [-i \hbar (\frac{\partial}{\partial x})]$$ are operators

### Observable

Location $$x$$ to find a particle has the PDF distribution. It is a valid distribution because of the normalization contraint.

$f_t(x) = |\Psi(x,t)|^2$

# Reference

[1] Sakurai, Jun John, and Eugene D. Commins. "Modern quantum mechanics, revised edition." (1995): 93-95.

[2] David J.Griffiths and Darrell F. Schroeter Introduction to Quantum Mechanics