Degrees of Freedom

A state $\ket{\psi}\in H$ of a is a vector in complex .

Generally a particle has multiple degrees of freedom

H=H_{space}\otimes H_{spin} \otimes ...

$H_{space}$ has continuous degrees of freedom -> infinite-dimensional $H_{spin}$ has discrete degrees of freedom -> finite-dimensional

E.g., electron in space or confined in a box or a quantum well

Finite Dimension Hilbert Space

Example $H_{spin}$ only has $\left\{\pm\frac{\hbar}{2}\right\}$ or $\{+\hbar, 0, -\hbar\}$ The spin can be described by

\ket{\psi_{spin}}=\sum_{n=0}^{d-1}\psi_n\ket{n}\quad\in\quad \mathbb{C}^d

Note that $\psi_n$ here is a Wave function Inner product Note the use of Cases

\braket{n|m}=\delta_{nm}=\begin{cases}0&n\neq m\\1&n=m\end{cases}

\braket{\phi_{spin}|\psi_{spin}}=\sum_{n=0}^{d-1}|\psi_n|^2

Normalization

1=\braket{\psi_{spin}|\psi_{spin}}=\sum_{n=0}^{d-1}|\psi_n|^2

Pr(n)=|\psi_n|^2

See Non Denumerable Basis Example $H_{space}$ has $x\in\mathbb{R}^d$

\ket{\psi_{space}}=\sum_{k=-N}^N\psi_k \ket{k}=\sum_{k=-N}^N\frac{\psi_k}{\sqrt{\Delta x}}\frac{\ket{k}}{\sqrt{\Delta x}} \Delta x

=\sum_{x_k=-L}^L\psi(x_k)\ket{x_k}\Delta x

Using Riemann sums from Integration

1 = \sum_k |\psi_k|^2 = \sum_k |\psi(x_k)|^2\,\Delta x \xrightarrow{\Delta x \to 0} \int |\psi(x)|^2\, dx

Note this is called a Wave function It's like a function that outputs the eigen"values" of the output

inner product

\braket{\phi_{space}|\psi_{space}}=\int \phi^*(x)\psi(x)dx

normalization

1=\braket{\psi_{space}|\psi_{space}}=\int|\psi(x)|^2dx

Measurement probability

Pr(x\in[a,b])=\int_a^b |\psi(x)|^2dx

Note that

\ket{\vec{x}}\neq\vec{x}

LHS: infinity dimension $H$ state vector RHS: finite-dim $\mathbb{R}$ $H$ state vector

\ket{\alpha \vec{x}}\neq \alpha\ket{\vec{x}}

\ket{-\vec{x}}\neq -\ket{\vec{x}}