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Chapter 2 • Theory & Derivations

Mathematical Formulation & Postulates of Quantum Mechanics

The axiomatic mathematical structure of quantum theory: abstract Hilbert space, Dirac bra-ket notation, linear Hermitian operators, eigenvalue equations, spectral decomposition, commutation algebra, generalized uncertainty relations, and measurement postulates.

§2.1 Hilbert Space and Dirac Bra-Ket Notation

Mathematical Foundation: Hilbert Space $\mathcal{H}$

In quantum mechanics, the state of a physical system is represented by a state vector $|\psi\rangle$ living in a complex linear vector space endowed with an inner product: a Hilbert Space $\mathcal{H}$.

A Hilbert space satisfies four core mathematical criteria:

1. Linearity: If $|\psi_1\rangle, |\psi_2\rangle \in \mathcal{H}$, then for any complex scalars $c_1, c_2 \in \mathbb{C}$, the linear superposition $c_1|\psi_1\rangle + c_2|\psi_2\rangle \in \mathcal{H}$.

2. Inner Product: For any two vectors $|\phi\rangle, |\psi\rangle \in \mathcal{H}$, there exists a complex scalar $\langle \phi | \psi \rangle$ satisfying:

  • Conjugate symmetry: $\langle \phi | \psi \rangle = \langle \psi | \phi \rangle^*$
  • Linearity in the second argument: $\langle \phi | c_1 \psi_1 + c_2 \psi_2 \rangle = c_1 \langle \phi | \psi_1 \rangle + c_2 \langle \phi | \psi_2 \rangle$
  • Positive-definiteness: $\langle \psi | \psi \rangle \ge 0$, and $\langle \psi | \psi \rangle = 0 \iff |\psi\rangle = 0$.

3. Cauchy Completeness: Every Cauchy sequence of vectors in $\mathcal{H}$ converges to an element within $\mathcal{H}$ under the norm $||\psi|| = \sqrt{\langle \psi | \psi \rangle}$.

4. Separability: The space possesses a countable dense subset, ensuring the existence of an orthonormal basis.

Dirac Notation: Kets, Bras, and Dual Space
  • Ket Vector $|\psi\rangle$: Represents a state vector in $\mathcal{H}$.
  • Bra Vector $\langle \phi |$: Represents a continuous linear functional in the dual space $\mathcal{H}^*$. By the Riesz Representation Theorem, every bra $\langle \phi |$ corresponds uniquely to a ket $|\phi\rangle$.
  • Wavefunction in Position Representation: The continuous spatial wavefunction $\psi(x)$ is simply the inner product projection of the abstract state vector $|\psi\rangle$ onto the continuous coordinate basis kets $|x\rangle$:
$$\psi(x) = \langle x | \psi \rangle, \qquad \psi^*(x) = \langle \psi | x \rangle$$

The inner product of two wavefunctions is given by:

$$\langle \phi | \psi \rangle = \int_{-\infty}^{+\infty} \langle \phi | x \rangle \langle x | \psi \rangle dx = \int_{-\infty}^{+\infty} \phi^*(x) \psi(x) dx$$

§2.2 Linear and Hermitian Operators

Linear Operators

An operator $\hat{A}$ maps kets to kets: $\hat{A}|\psi\rangle = |\psi'\rangle$. An operator is linear if:

$$\hat{A}(c_1 |\psi_1\rangle + c_2 |\psi_2\rangle) = c_1 \hat{A}|\psi_1\rangle + c_2 \hat{A}|\psi_2\rangle$$
Adjoint (Hermitian Conjugate) Operator $\hat{A}^\dagger$

The adjoint of an operator $\hat{A}$, denoted $\hat{A}^\dagger$, is defined through the inner product relation:

$$\langle \phi | \hat{A} | \psi \rangle = \langle \hat{A}^\dagger \phi | \psi \rangle = \left( \langle \psi | \hat{A}^\dagger | \phi \rangle \right)^*$$

Properties of the adjoint:

  • $(\hat{A}^\dagger)^\dagger = \hat{A}$
  • $(c \hat{A})^\dagger = c^* \hat{A}^\dagger$
  • $(\hat{A} + \hat{B})^\dagger = \hat{A}^\dagger + \hat{B}^\dagger$
  • $(\hat{A}\hat{B})^\dagger = \hat{B}^\dagger \hat{A}^\dagger$
Hermitian (Self-Adjoint) Operators

An operator is Hermitian if it equals its own adjoint:

$$\hat{A}^\dagger = \hat{A} \iff \langle \phi | \hat{A} | \psi \rangle = \langle \hat{A} \phi | \psi \rangle$$
Fundamental Theorems of Hermitian Operators

1. Theorem 1: The eigenvalues of a Hermitian operator are strictly real.

  • Proof: Let $\hat{A}|\psi\rangle = a|\psi\rangle$ with $\langle \psi | \psi \rangle \ne 0$.
$$\langle \psi | \hat{A} | \psi \rangle = a \langle \psi | \psi \rangle$$

Taking the complex conjugate and using $\hat{A} = \hat{A}^\dagger$:

$$\langle \psi | \hat{A} | \psi \rangle^* = \langle \psi | \hat{A}^\dagger | \psi \rangle = \langle \psi | \hat{A} | \psi \rangle = a^* \langle \psi | \psi \rangle$$

Therefore, $(a - a^)\langle \psi | \psi \rangle = 0$. Since $\langle \psi | \psi \rangle > 0$, we have $a = a^ \implies a \in \mathbb{R}$.

2. Theorem 2: Eigenvectors corresponding to distinct eigenvalues are orthogonal.

  • Proof: Let $\hat{A}|\phi_1\rangle = a_1|\phi_1\rangle$ and $\hat{A}|\phi_2\rangle = a_2|\phi_2\rangle$ with $a_1 \ne a_2$.
$$\langle \phi_2 | \hat{A} | \phi_1 \rangle = a_1 \langle \phi_2 | \phi_1 \rangle$$
$$\langle \phi_2 | \hat{A} | \phi_1 \rangle = \langle \hat{A} \phi_2 | \phi_1 \rangle = a_2^* \langle \phi_2 | \phi_1 \rangle = a_2 \langle \phi_2 | \phi_1 \rangle$$

Subtracting gives: $(a_1 - a_2) \langle \phi_2 | \phi_1 \rangle = 0$. Since $a_1 \ne a_2$, it follows that $\langle \phi_2 | \phi_1 \rangle = 0$.

§2.3 Commutator Algebra and Compatible Observables

Commutator Definition and Properties

The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as:

$$[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$$

Fundamental identities:

  • Anti-symmetry: $[\hat{A}, \hat{B}] = -[\hat{B}, \hat{A}]$
  • Linearity: $[\hat{A}, \hat{B} + \hat{C}] = [\hat{A}, \hat{B}] + [\hat{A}, \hat{C}]$
  • Product rule: $[\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}]\hat{C} + \hat{B}[\hat{A}, \hat{C}]$
  • Product rule: $[\hat{A}\hat{B}, \hat{C}] = \hat{A}[\hat{B}, \hat{C}] + [\hat{A}, \hat{C}]\hat{B}$
  • Jacobi Identity: $[\hat{A}, [\hat{B}, \hat{C}]] + [\hat{B}, [\hat{C}, \hat{A}]] + [\hat{C}, [\hat{A}, \hat{B}]] = 0$
The Fundamental Position-Momentum Commutator

Applying $[\hat{x}, \hat{p}]$ to an arbitrary differentiable test function $f(x)$:

$$[\hat{x}, \hat{p}]f(x) = x\left(-i\hbar \frac{df}{dx}\right) - \left(-i\hbar \frac{d}{dx}(x f(x))\right) = -i\hbar x \frac{df}{dx} + i\hbar\left(f + x\frac{df}{dx}\right) = i\hbar f(x)$$
$$\implies [\hat{x}, \hat{p}] = i\hbar \hat{I}$$
Generalized Robertson-Schrödinger Uncertainty Relation

For any two Hermitian observables $\hat{A}$ and $\hat{B}$:

$$\Delta A \cdot \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|$$

If $[\hat{A}, \hat{B}] = 0$, the observables are compatible: they share a common complete set of simultaneous eigenstates and can be measured simultaneously to arbitrary precision. If $[\hat{A}, \hat{B}] \ne 0$, they are incompatible, giving rise to an uncertainty relation.

§2.4 The Postulates of Quantum Mechanics

The complete theoretical structure of nonrelativistic quantum mechanics is founded upon five core postulates:

1. Postulate 1 (State of the System): At any given time $t$, the state of a physical system is completely specified by a normalized state vector $|\psi(t)\rangle$ residing in a complex Hilbert space $\mathcal{H}$.

2. Postulate 2 (Observables): Every physically measurable dynamical variable $\mathcal{A}$ is represented by a linear Hermitian operator $\hat{A}$ acting in $\mathcal{H}$.

3. Postulate 3 (Possible Measurement Outcomes): The measurement of an observable $\mathcal{A}$ can yield only one of the eigenvalues $a_n$ of the corresponding Hermitian operator equation:

$$\hat{A}|\phi_n\rangle = a_n |\phi_n\rangle$$

4. Postulate 4 (Born's Probabilistic Interpretation & State Reduction): If a system is in state $|\psi\rangle$, the probability of obtaining non-degenerate eigenvalue $a_n$ upon measurement of $\hat{A}$ is given by:

$$P(a_n) = \frac{|\langle \phi_n | \psi \rangle|^2}{\langle \psi | \psi \rangle}$$

Immediately following the measurement, if eigenvalue $a_n$ is obtained, the state vector collapses into the corresponding eigenstate:

$$|\psi\rangle \xrightarrow{\text{measurement}} |\phi_n\rangle$$

The expectation value of $\hat{A}$ across an ensemble of identically prepared systems is:

$$\langle \hat{A} \rangle = \frac{\langle \psi | \hat{A} | \psi \rangle}{\langle \psi | \psi \rangle}$$

5. Postulate 5 (Time Evolution): Between measurements, the time evolution of the state vector $|\psi(t)\rangle$ is governed by the deterministic Time-Dependent Schrödinger Equation:

$$i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}(t)|\psi(t)\rangle$$

where $\hat{H}$ is the Hamiltonian operator of the system.

Examine state reduction in Simulation 2.1 below, which lets you prepare a two-level superposition state and trigger wavefunction collapse into individual eigenstates.

📝 Chapter Worked Examples & Exercises

Complete derivations & analytical proofs
Hard Example 2.1: Commutator Algebra: [x^n, p] and Generalized Ehrenfest Relation

Prove by induction that $[\hat{x}^n, \hat{p}] = i\hbar n \hat{x}^{n-1}$ for any integer $n \ge 1$. Hence, evaluate $[f(\hat{x}), \hat{p}]$ for any analytic function $f(x)$.

Step 1: Base Case and Product Rule Induction
$$\text{For } n = 1: [\hat{x}, \hat{p}] = i\hbar \hat{I} \quad (\text{Holds}) $$ $$\text{Assume true for } n = k: [\hat{x}^k, \hat{p}] = i\hbar k \hat{x}^{k-1} $$ $$\text{For } n = k+1: [\hat{x}^{k+1}, \hat{p}] = [\hat{x}^k \hat{x}, \hat{p}] = \hat{x}^k [\hat{x}, \hat{p}] + [\hat{x}^k, \hat{p}]\hat{x} $$ $$= \hat{x}^k (i\hbar) + (i\hbar k \hat{x}^{k-1})\hat{x} = i\hbar \hat{x}^k + i\hbar k \hat{x}^k = i\hbar (k+1) \hat{x}^k$$

The relation holds for $n=1$, and if it holds for $n=k$, it holds for $n=k+1$. By mathematical induction, it holds for all positive integers $n$.

Step 2: Extension to Analytic Functions
$$f(\hat{x}) = \sum_{n=0}^\infty c_n \hat{x}^n $$ $$[f(\hat{x}), \hat{p}] = \sum_{n=0}^\infty c_n [\hat{x}^n, \hat{p}] = \sum_{n=1}^\infty c_n (i\hbar n \hat{x}^{n-1}) = i\hbar \frac{df(\hat{x})}{d\hat{x}}$$

This commutator identity is widely used in quantum mechanics to derive the equations of motion for expectation values.

An electron is accelerated from rest through an electrostatic potential difference of $V = 150 \text{ V}$.\n(a) Determine its de Broglie wavelength using non-relativistic mechanics.\n(b) At what accelerating potential does the relativistic correction to the de Broglie wavelength exceed $1\%$?
Step 1: Non-relativistic Calculation
$$K = e V = 150 \text{ eV} = 150 \times 1.602 \times 10^{-19} \text{ J} = 2.403 \times 10^{-17} \text{ J} $$ $$p = \sqrt{2 m_e K} = \sqrt{2(9.109 \times 10^{-31} \text{ kg})(2.403 \times 10^{-17} \text{ J})} = 6.617 \times 10^{-24} \text{ kg}\cdot\text{m/s} $$ $$\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34} \text{ J}\cdot\text{s}}{6.617 \times 10^{-24} \text{ kg}\cdot\text{m/s}} = 1.001 \times 10^{-10} \text{ m} = 1.001 \text{ Å}$$
For quick calculations, note that $\lambda = \sqrt{\frac{150}{V}} \text{ Å}$. For $V = 150 \text{ V}$, $\lambda = 1.00 \text{ Å}$, which corresponds to typical atomic crystal lattice spacings.
Step 2: Relativistic Condition
$$E^2 = p^2 c^2 + m_0^2 c^4 \implies p = \frac{1}{c}\sqrt{K(K + 2m_0 c^2)} $$ $$\lambda_{\text{rel}} = \frac{h c}{\sqrt{K(K + 2m_0 c^2)}} = \frac{\lambda_{\text{class}}}{\sqrt{1 + \frac{K}{2m_0 c^2}}} \approx \lambda_{\text{class}}\left(1 - \frac{K}{4 m_0 c^2}\right) $$ $$\frac{\Delta \lambda}{\lambda} \approx \frac{K}{4 m_0 c^2} \ge 0.01 \implies K \ge 0.04 m_0 c^2 = 0.04 (511 \text{ keV}) \approx 20.44 \text{ keV}$$
When accelerating potentials exceed roughly $20 \text{ kV}$ (typical in transmission electron microscopes), relativistic momentum corrections become necessary.