Quantum Mechanics: Postulates, Operators, and System Classification

Classification of Physical Systems: Classical vs. Quantum

Heisenberg's uncertainty relations provide the criteria for determining whether a physical system should be treated as classical or quantum. This distinction is not based solely on a qualitative assessment of spatial extension, but rather on comparing the characteristic action (SS) of the system with Planck's constant (ℏ\hbar).

Criteria for System Type:

  • A system is judged based on its characteristic action defined by the product of its spatial dimensions and momentum (S≈d×pS \approx d \times p).
  • Quantum System: When the action is of the order of Planck's constant (S≈ℏS \approx \hbar).
  • Classical System: When the action is much larger than Planck's constant (S≫ℏS \gg \hbar).

Case Studies and Examples:

  1. Electron in a Hydrogen Atom:

    • Atomic Diameter (dd): ≈10−10 m\approx 10^{-10}\,m
    • Electron Momentum (pp) on a Bohr orbit: ≈10−24 kg m s−1\approx 10^{-24}\,kg\,m\,s^{-1}
    • Action (SS): ≈10−10×10−24=10−34 Js\approx 10^{-10} \times 10^{-24} = 10^{-34}\,Js
    • Comparison: Since S≈ℏ≈10−34 JsS \approx \hbar \approx 10^{-34}\,Js, the electron in an atom must be treated as a quantum system.

  2. Electron in a Electronic Tube (CRT):

    • Electrode Distance (dd): ≈10−1 m\approx 10^{-1}\,m
    • Potential Difference (VV): ≈103 V\approx 10^3\,V
    • Electron Momentum (pp): ≈1.7×10−24 kg m s−1\approx 1.7 \times 10^{-24}\,kg\,m\,s^{-1}
    • Action (SS): ≈1.7×10−25 Js\approx 1.7 \times 10^{-25}\,Js
    • Comparison: Since S≫ℏS \gg \hbar, electrons in electronic tubes can be treated as classical systems.

  3. Electron in a Tunnel Diode:

    • Voltage: 10 V10\,V
    • Momentum (pp): ≈1.7×10−24 kg m s−1\approx 1.7 \times 10^{-24}\,kg\,m\,s^{-1}
    • Potential Barrier Width (dd): ≈10−8 m\approx 10^{-8}\,m
    • Action (SS): ≈1.7×10−32 Js\approx 1.7 \times 10^{-32}\,Js
    • Comparison: Since the action is close to ℏ\hbar, the electron behaves as a quantum system.

Classical systems are governed by the laws of classical physics, while quantum systems are governed by the postulates of quantum mechanics.

The First Postulate of Quantum Mechanics: Observables and Operators

In any physical theory, fundamental quantities are the observables of the system—entities that can be measured through reproducible operations. While classical physics treats observables as functions of basic variables (like coordinates and momenta), quantum mechanics introduces a new mathematical representation.

The First Postulate States: To every physical observable AA of a quantum system, there corresponds a linear Hermitian operator A^\hat{A} in Hilbert space. The various eigenvalues of this operator are the measurable values of the observable it represents.

Mathematical Implications:

  • In quantum physics, dynamic observables are associated with operators acting on the state of the system.
  • The notion of an observable is strictly tied to the measurement process.
  • The requirement for the operator to be Hermitian ensures that all its eigenvalues are real numbers, matching physical reality where measurements yield real results.

Hermiticity Condition: For any wave function or vector in Hilbert space ψ\psi, it is necessary that: A^ψ=aψ\hat{A} \psi = a \psi More restrictively, for an operator to be an observable, for any ψ\psi and ϕ\phi, the following must hold: ∫ϕ∗A^ψ dx=∫(A^ϕ)∗ψ dx\int \phi^* \hat{A} \psi \, dx = \int (\hat{A} \phi)^* \psi \, dx Alternatively expressed as: ⟨ϕ∣A^∣ψ⟩=⟨A^ϕ∣ψ⟩\langle \phi | \hat{A} | \psi \rangle = \langle \hat{A} \phi | \psi \rangle

Mathematical Consequences for Quantum Observables

The mathematical nature of Hermitian operators leads to several critical physical consequences utilized in applications:

  1. Real Eigenvalues:     If ana_n are the eigenvalues of A^\hat{A}, then:     A^ψn=anψn\hat{A} \psi_n = a_n \psi_n     Multiplying by ψn∗\psi_n^* and integrating leads to:     ∫ψn∗A^ψn dx=an∫ψn∗ψn dx\int \psi_n^* \hat{A} \psi_n \, dx = a_n \int \psi_n^* \psi_n \, dx     Since both sides and the integral of the probability density are real, ana_n must be real.

  2. Completeness (Closure Relation):     The eigenvectors ψn\psi_n of an observable form a complete set (basis). Any arbitrary state vector ψ(x)\psi(x) can be expressed as a linear combination of these eigenvectors:     ψ(x)=∑nCnψn(x)\psi(x) = \sum_n C_n \psi_n(x)     In the case of a continuous spectrum of eigenvalues:     ψ(x)=∫C(a)ψa(x) da\psi(x) = \int C(a) \psi_a(x) \, da

  3. Orthogonality of Eigenvectors:     Eigenvectors corresponding to different eigenvalues are orthogonal (their overlap integral is zero). If a1≠a2a_1 \neq a_2:     ∫ψ1∗ψ2 dx=0\int \psi_1^* \psi_2 \, dx = 0     This demonstrates that distinct physical states of an observable are perfectly distinguishable mathematically.

The Second Postulate: The Correspondence Principle and Quantization

The second postulate addresses how to assign a specific operator to a known physical quantity.

The Second Postulate States: Any classical physical quantity can be considered as constructed from pairs of canonically conjugate variables. The linear Hermitian operator corresponding to such a dynamic observable is obtained by replacing the classical canonical variables with their corresponding operators.

Heisenberg Commutation Rules: For all pairs of operators representing basic canonical variables, the following quantization axioms apply: [q^k,q^l]=0[\hat{q}_k, \hat{q}_l] = 0[p^k,p^l]=0[\hat{p}_k, \hat{p}_l] = 0[p^k,q^l]=−iℏδkl[\hat{p}_k, \hat{q}_l] = -i\hbar \delta_{kl}

Notes on the Second Postulate:

  • If a system has no classical correspondent (e.g., spin, isobaric spin), the first part of this postulate is not directly applicable. In those cases, ad-hoc methods, symmetry properties, or conservation rules are used to define the operators.
  • The description of a quantum system in terms of q^\hat{q} and p^\hat{p} is considered complete.
  • Operators satisfying the commutation relations are related by unitary transformations:   q^l′=Uq^lU−1\hat{q}'_l = U \hat{q}_l U^{-1}p^l′=Up^lU−1\hat{p}'_l = U \hat{p}_l U^{-1}

Examples of Operators and Commutators

1. Kinetic Moment (Angular Momentum): For a particle, the classical angular momentum is l=r×pl = r \times p. The quantum components are: l^x=y^p^z−z^p^y\hat{l}_x = \hat{y} \hat{p}_z - \hat{z} \hat{p}_yl^y=z^p^x−x^p^z\hat{l}_y = \hat{z} \hat{p}_x - \hat{x} \hat{p}_zl^z=x^p^y−y^p^x\hat{l}_z = \hat{x} \hat{p}_y - \hat{y} \hat{p}_x Total angular momentum squared: l^2=l^x2+l^y2+l^z2\hat{l}^2 = \hat{l}_x^2 + \hat{l}_y^2 + \hat{l}_z^2

Angular Momentum Commutation Relations:[l^x,l^y]=iℏl^z[\hat{l}_x, \hat{l}_y] = i\hbar \hat{l}_z[l^y,l^z]=iℏl^x[\hat{l}_y, \hat{l}_z] = i\hbar \hat{l}_x[l^z,l^x]=iℏl^y[\hat{l}_z, \hat{l}_x] = i\hbar \hat{l}_y Relationship with the magnitude: [l^2,l^x]=[l^2,l^y]=[l^2,l^z]=0[\hat{l}^2, \hat{l}_x] = [\hat{l}^2, \hat{l}_y] = [\hat{l}^2, \hat{l}_z] = 0

2. One-Dimensional Harmonic Oscillator: A particle of mass mm in an elastic potential has a Hamiltonian: H=p22m+12kx2⇒H^=p^22m+12kx^2H = \frac{p^2}{2m} + \frac{1}{2} k x^2 \Rightarrow \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2} k \hat{x}^2

Commutator Calculation Rules: To calculate complex commutators where AA is an observable represented as a power series in q^\hat{q} or p^\hat{p}, the following identities are used:

  1. [p^,q^n]=−iℏnq^n−1[\hat{p}, \hat{q}^n] = -i\hbar n \hat{q}^{n-1}
  2. [q^,p^n]=iℏnp^n−1[\hat{q}, \hat{p}^n] = i\hbar n \hat{p}^{n-1}
  3. [p^,A(q^)]=−iℏ∂A∂q[\hat{p}, A(\hat{q})] = -i\hbar \frac{\partial A}{\partial q}
  4. [q^,A(p^)]=iℏ∂A∂p[\hat{q}, A(\hat{p})] = i\hbar \frac{\partial A}{\partial p}