Quantum Foundations

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

States

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

$$\bra{a} = (a_1, a_2, \ldots, a_n),\qquad \ket{b} = \begin{pmatrix}b_1\\b_2\\ \vdots \\ b_n\end{pmatrix}.$$

This Dirac notation elegantly captures the structure of quantum states.

An operator $\bold M$ acting on a ket satisfies

$$\bold M\ket{\lambda} = \lambda\,\ket{\lambda},$$

where $\lambda$ is an eigenvalue and $\ket{\lambda}$ is its eigenvector.

Observability

Observables correspond to Hermitian operators:

$$\bold L = \bold L^{\dagger}.$$

For a Hermitian operator,

$$\bold L\ket{\lambda} = \lambda\ket{\lambda}, \quad \bra{\lambda}\bold L^{\dagger} = \lambda^*\bra{\lambda}.$$

Then

$$\bra{\lambda}\bold L\ket{\lambda} = \lambda\,\braket{\lambda|\lambda} = \bra{\lambda}\bold L^{\dagger}\ket{\lambda} = \lambda^*\,\braket{\lambda|\lambda}.$$

Since $\bold L=\bold L^{\dagger}$ and $\braket{\lambda|\lambda}\neq0$, it follows that $\lambda=\lambda^*$; hence, observables yield real values.

Observable and measurable quantities are represented by Hermitian operators.

Measurements

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

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

$$\ket{\Psi(t)} = \bold U(t)\ket{\Psi(0)}, \qquad \bold U^{\dagger}(t)\,\bold U(t) = I.$$

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

Classical	State and measurement coincide.
Quantum	State evolution (via $\bold U$) and measurement (via $\bold L$) are distinct.

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

$$P(\lambda_i) = \braket{A|\lambda_i}\,\braket{\lambda_i|A}\,.$$

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

Energy and Time Evolution

For an infinitesimal interval $\epsilon$, unitarity requires

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

Expanding to first order,

$$\bold U(\epsilon) = I - i\epsilon\,\bold H, \quad \bold U^{\dagger}(\epsilon) = I + i\epsilon\,\bold H^{\dagger}.$$

Applying this to the state,

$$\ket{\Psi(\epsilon)} = (I - i\epsilon\,\bold H)\ket{\Psi(0)},$$

leads to

$$\frac{\ket{\Psi(\epsilon)} - \ket{\Psi(0)}}{\epsilon} = -i\,\bold H\,\ket{\Psi(0)},$$

which in the limit $\epsilon\to0$ becomes the time‑dependent Schrödinger equation:

$$i\hbar\frac{\partial}{\partial t}\ket{\Psi(t)} = \bold H\,\ket{\Psi(t)}.$$

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