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