Quantum Foundations

It's ok—they're just vectors. Written 9 months ago.

The fundamental principles of quantum mechanics are stored in quote blocks.

States

Quantum states live in vector spaces, represented by bras and kets:

⟨a∣=(a1,a2,…,an),∣b⟩=(b1b2⋮bn).\bra{a} = (a_1, a_2, \ldots, a_n),\qquad \ket{b} = \begin{pmatrix}b_1\\b_2\\ \vdots \\ b_n\end{pmatrix}.

This Dirac notation elegantly captures the structure of quantum states.

An operator M\bold M acting on a ket satisfies

M∣λ⟩=λ ∣λ⟩,\bold M\ket{\lambda} = \lambda\,\ket{\lambda},

where λ\lambda is an eigenvalue and ∣λ⟩\ket{\lambda} is its eigenvector.

Observability

Observables correspond to Hermitian operators:

L=L†.\bold L = \bold L^{\dagger}.

For a Hermitian operator,

L∣λ⟩=λ∣λ⟩,⟨λ∣L†=λ∗⟨λ∣.\bold L\ket{\lambda} = \lambda\ket{\lambda}, \quad \bra{\lambda}\bold L^{\dagger} = \lambda^*\bra{\lambda}.

Then

⟨λ∣L∣λ⟩=λ ⟨λ∣λ⟩=⟨λ∣L†∣λ⟩=λ∗ ⟨λ∣λ⟩.\bra{\lambda}\bold L\ket{\lambda} = \lambda\,\braket{\lambda|\lambda} = \bra{\lambda}\bold L^{\dagger}\ket{\lambda} = \lambda^*\,\braket{\lambda|\lambda}.

Since L=L†\bold L=\bold L^{\dagger} and ⟨λ∣λ⟩≠0\braket{\lambda|\lambda}\neq0, it follows that λ=λ∗\lambda=\lambda^*; hence, observables yield real values.

Observable and measurable quantities are represented by Hermitian operators.

Measurements

The possible outcomes of measuring an observable L\bold L are its eigenvalues {λi}\{\lambda_i\}. If the system is in eigenstate ∣λi⟩\ket{\lambda_i}, the measurement result is guaranteed to be λi\lambda_i.

Given an initial state ∣Ψ(0)⟩\ket{\Psi(0)}, its time evolution is governed by a unitary operator U(t)\bold U(t):

∣Ψ(t)⟩=U(t)∣Ψ(0)⟩,U†(t) U(t)=I.\ket{\Psi(t)} = \bold U(t)\ket{\Psi(0)}, \qquad \bold U^{\dagger}(t)\,\bold U(t) = I.

Unitarity preserves inner products, separating quantum dynamics from classical intuition:

ClassicalState and measurement coincide.
QuantumState evolution (via U\bold U) and measurement (via L\bold L) are distinct.

After evolving to ∣A⟩\ket{A}, measuring L\bold L yields probabilities

P(λi)=⟨A∣λi⟩ ⟨λi∣A⟩ .P(\lambda_i) = \braket{A|\lambda_i}\,\braket{\lambda_i|A}\,.

If ∣A⟩\ket{A} is the post-evolution state, the probability of obtaining λi\lambda_i upon measuring L\bold L is P(λi)=⟨A∣λi⟩⟨λi∣A⟩P(\lambda_i)=\braket{A|\lambda_i}\braket{\lambda_i|A}.

Energy and Time Evolution

For an infinitesimal interval ϵ\epsilon, unitarity requires

U†(ϵ) U(ϵ)=I.\bold U^{\dagger}(\epsilon)\,\bold U(\epsilon) = I.

Expanding to first order,

U(ϵ)=I−iϵ H,U†(ϵ)=I+iϵ H†.\bold U(\epsilon) = I - i\epsilon\,\bold H, \quad \bold U^{\dagger}(\epsilon) = I + i\epsilon\,\bold H^{\dagger}.

Applying this to the state,

∣Ψ(ϵ)⟩=(I−iϵ H)∣Ψ(0)⟩,\ket{\Psi(\epsilon)} = (I - i\epsilon\,\bold H)\ket{\Psi(0)},

leads to

∣Ψ(ϵ)⟩−∣Ψ(0)⟩ϵ=−i H ∣Ψ(0)⟩,\frac{\ket{\Psi(\epsilon)} - \ket{\Psi(0)}}{\epsilon} = -i\,\bold H\,\ket{\Psi(0)},

which in the limit ϵ→0\epsilon\to0 becomes the time‑dependent Schrödinger equation:

iℏ∂∂t∣Ψ(t)⟩=H ∣Ψ(t)⟩.i\hbar\frac{\partial}{\partial t}\ket{\Psi(t)} = \bold H\,\ket{\Psi(t)}.