Degrees of Freedom

A state ∣ψ⟩∈H\ket{\psi}\in H of a is a vector in complex .

Generally a particle has multiple degrees of freedom

H=Hspace⊗Hspin⊗...H=H_{space}\otimes H_{spin} \otimes ...

HspaceH_{space} has continuous degrees of freedom -> infinite-dimensional HspinH_{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 HspinH_{spin} only has {±ℏ2}\left\{\pm\frac{\hbar}{2}\right\} or {+ℏ,0,−ℏ}\{+\hbar, 0, -\hbar\} The spin can be described by

∣ψspin⟩=∑n=0d−1ψn∣n⟩∈Cd\ket{\psi_{spin}}=\sum_{n=0}^{d-1}\psi_n\ket{n}\quad\in\quad \mathbb{C}^d

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

⟨n∣m⟩=δnm={0n≠m1n=m\braket{n|m}=\delta_{nm}=\begin{cases}0&n\neq m\\1&n=m\end{cases} ⟨ϕspin∣ψspin⟩=∑n=0d−1∣ψn∣2\braket{\phi_{spin}|\psi_{spin}}=\sum_{n=0}^{d-1}|\psi_n|^2

Normalization

1=⟨ψspin∣ψspin⟩=∑n=0d−1∣ψn∣21=\braket{\psi_{spin}|\psi_{spin}}=\sum_{n=0}^{d-1}|\psi_n|^2

Measurement probability: Born rule

Pr(n)=∣ψn∣2Pr(n)=|\psi_n|^2

Infinite Dimension Hilbert Space

See Non Denumerable Basis Example HspaceH_{space} has x∈Rdx\in\mathbb{R}^d

∣ψspace⟩=∑k=−NNψk∣k⟩=∑k=−NNψkΔx∣k⟩ΔxΔx\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 =∑xk=−LLψ(xk)∣xk⟩Δx=\sum_{x_k=-L}^L\psi(x_k)\ket{x_k}\Delta x

Using Riemann sums from Integration

1=∑k∣ψk∣2=∑k∣ψ(xk)∣2 Δx→Δx→0∫∣ψ(x)∣2 dx1 = \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

⟨ϕspace∣ψspace⟩=∫ϕ∗(x)ψ(x)dx\braket{\phi_{space}|\psi_{space}}=\int \phi^*(x)\psi(x)dx

normalization

1=⟨ψspace∣ψspace⟩=∫∣ψ(x)∣2dx1=\braket{\psi_{space}|\psi_{space}}=\int|\psi(x)|^2dx

Measurement probability

Pr(x∈[a,b])=∫ab∣ψ(x)∣2dxPr(x\in[a,b])=\int_a^b |\psi(x)|^2dx

See Probability Density

Note

Note that

∣x⃗⟩≠x⃗\ket{\vec{x}}\neq\vec{x}

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

∣αx⃗⟩≠α∣x⃗⟩\ket{\alpha \vec{x}}\neq \alpha\ket{\vec{x}} ∣−x⃗⟩≠−∣x⃗⟩\ket{-\vec{x}}\neq -\ket{\vec{x}}