10.5 Postulates of Quantum Mechanics, Operators, Particle in a Box, and Harmonic Oscillator

Postulates of Quantum Mechanics, Operators, Particle in a Box, and Harmonic Oscillator

At the close of the nineteenth century, classical Newtonian mechanics and Maxwellian electrodynamics failed completely when applied to subatomic phenomena and radiation-matter interactions. The emergence of quantum mechanics introduced an entirely new physical paradigm where energy, angular momentum, and physical observables are quantized, matter exhibits wave-particle duality, and physical states are described probabilistically by wave functions operating under linear Hermitian operators.

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1. Classical Failures and Early Quantum Milestones

1. Blackbody Radiation and Planck's Quantum Hypothesis (1900)

Classical Rayleigh-Jeans theory modeled cavity radiation as electromagnetic standing waves in thermal equilibrium, predicting that spectral energy density $d\rho$ increases with the inverse fourth power of wavelength: $d\rho = \frac{8\pi k_B T}{\lambda^4} d\lambda$ As $\lambda \to 0$ (ultraviolet region), this classical formula predicted that radiated energy density diverges to infinity—an absurd physical catastrophe known as the ultraviolet catastrophe.

Max Planck solved this by postulating that matter absorbs and emits radiation only in discrete packets or quanta of energy proportional to radiation frequency $\nu$: $E = n h \nu \quad (n = 0, 1, 2, \dots)$ where $h = 6.62607 \times 10^{-34}\text{ J}\cdot\text{s}$ is Planck's constant. Planck's radiation law reproduced experimental blackbody spectra across all wavelengths: $\rho(\nu, T) d\nu = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu / k_B T} - 1} d\nu$

2. The Photoelectric Effect (Einstein, 1905)

When monochromatic light strikes a clean metal surface in a vacuum, electrons are ejected immediately, but only if the light frequency exceeds a critical threshold frequency ($\nu_0$):

• Classical wave theory predicted that electron kinetic energy should increase with light intensity (wave amplitude), and that dim light would require an accumulation time delay.
• Experimentally, light intensity alters only the number of ejected photoelectrons (photocurrent), while the maximum kinetic energy depends strictly on light frequency.

Albert Einstein resolved this by proposing that electromagnetic radiation consists of localized energy packets called photons, each bearing energy $E = h\nu$. Energy conservation requires:

$E_k = h\nu - \Phi = h\nu - h\nu_0$ where $\Phi = h\nu_0$ is the work function of the metal (the minimum binding energy required to liberate an electron from the Fermi level into the vacuum).

3. Compton Scattering (1923)

Arthur Compton bombarded graphite targets with monochromatic X-rays and observed that scattered X-rays had a longer wavelength ($\lambda' > \lambda$) than incident rays. Treating the collision as a relativistic elastic impact between a photon carrying momentum $p = h/\lambda$ and a stationary electron yields the Compton wavelength shift:

$\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c}(1 - \cos\theta)$ where $\frac{h}{m_e c} = 2.4263 \times 10^{-12}\text{ m} = 0.02426\text{ \AA}$ is the Compton wavelength of an electron, and $\theta$ is the scattering angle.

4. De Broglie Matter Waves (1924)

Louis de Broglie hypothesized that if light waves possess particle characteristics (momentum $p$), then material particles must possess wave characteristics. The de Broglie wavelength of any particle with linear momentum $p = mv$ is:

$\lambda = \frac{h}{p} = \frac{h}{mv}$ Electron wave properties were confirmed experimentally in 1927 by Clinton Davisson and Lester Germer (observing electron diffraction from nickel crystals) and George Thomson (thin gold foil transmission).

5. Heisenberg Uncertainty Principle (1927)

Werner Heisenberg proved that due to wave-particle duality, it is physically impossible to measure simultaneously both the position and conjugate linear momentum of a quantum entity with arbitrary precision:

$\Delta x \Delta p_x \ge \frac{\hbar}{2}$ where $\hbar \equiv \frac{h}{2\pi} = 1.05457 \times 10^{-34}\text{ J}\cdot\text{s}$. A parallel uncertainty relation links energy and time: $\Delta E \Delta t \ge \frac{\hbar}{2}$.

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2. The Five Postulates of Quantum Mechanics

Modern quantum theory is constructed axiomatically upon five foundational postulates:

Postulate 1: The State and the Wave Function

The physical state of a system of particles is completely specified by a continuous, single-valued complex wave function $\Psi(\mathbf{r}_1, \mathbf{r}_2, \dots, t)$.

• Born Interpretation (Max Born, 1926): The wave function itself carries no direct classical physical dimension, but the product of $\Psi$ with its complex conjugate $\Psi^*$—the squared modulus $|\Psi|^2$—represents the probability density: $P(x, y, z) d\tau = |\Psi|^2 d\tau = \Psi^* \Psi\,d\tau$ representing the probability of locating the particle within spatial volume element $d\tau = dx\,dy\,dz$.
• Well-Behaved Criteria: For $\Psi$ to represent a physically realizable state, it must satisfy three conditions everywhere:
  1. $\Psi$ must be single-valued ($f(x)$ has only one value at each point).
  2. $\Psi$ and its first spatial derivatives ($\frac{d\Psi}{dx}$) must be continuous.
  3. $\Psi$ must be quadratically integrable (normalizable), so the total probability over all space equals unity: $\int_{-\infty}^{+\infty} |\Psi|^2 d\tau = 1$

Postulate 2: Physical Observables and Operators

To every physically observable quantity in classical mechanics (position, momentum, kinetic energy, angular momentum) corresponds a linear Hermitian operator in quantum mechanics.

Here, $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ is the Laplacian operator.

Postulate 3: Eigenvalues and Measurement

If a physical observable represented by operator $\hat{A}$ is measured on a system, the only possible numerical outcome of the measurement is one of the eigenvalues ($a_n$) that satisfy the eigenvalue equation:

$\hat{A} \psi_n = a_n \psi_n$ where $\psi_n$ is the corresponding eigenfunction.

• If the system is already in an eigenstate $\psi_n$, measuring observable $A$ yields eigenvalue $a_n$ with 100% certainty.
• If the system is in an arbitrary state $\Psi$ (a linear combination $\Psi = \sum c_n \psi_n$), measuring $A$ yields eigenvalue $a_n$ with probability $|c_n|^2$, and the measurement instantly collapses the wave function to eigenstate $\psi_n$.

Postulate 4: Expectation Values

When an observable $A$ is measured repeatedly on an ensemble of identically prepared quantum systems described by normalized wave function $\Psi$, the statistical average or expectation value ($\langle A \rangle$) is:

$\langle A \rangle = \int_{-\infty}^{+\infty} \Psi^* \hat{A} \Psi\,d\tau$

Postulate 5: Time Evolution of the State

The time-dependent evolution of a wave function is governed by the Time-Dependent Schrödinger Equation (TDSE):

$i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi$ When the potential energy $V(\mathbf{r})$ is independent of time, separation of variables $\Psi(\mathbf{r}, t) = \psi(\mathbf{r}) e^{-iEt / \hbar}$ yields the stationary Time-Independent Schrödinger Equation (TISE):

$\hat{H} \psi = E \psi \iff -\frac{\hbar^2}{2m}\nabla^2 \psi + V(\mathbf{r})\psi = E\psi$

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3. Mathematical Properties of Operators: Hermitian and Commutator Algebra

Hermitian Operators

An operator $\hat{A}$ is defined as Hermitian if it satisfies the integral equality for all well-behaved wave functions $f$ and $g$: $\int f^* (\hat{A} g)\,d\tau = \int (\hat{A} f)^* g\,d\tau$

Two Essential Theorems:

1. Real Eigenvalues: All eigenvalues of a Hermitian operator are strictly real numbers ($a_n \in \mathbb{R}$). Because physical measurements in the laboratory yield real quantities, all quantum operators representing observables must be Hermitian.
2. Orthogonality of Eigenfunctions: Eigenfunctions of a Hermitian operator corresponding to distinct eigenvalues are mutually orthogonal: $\int \psi_m^* \psi_n\,d\tau = 0 \quad (\text{for } a_m \ne a_n)$ Combining normalization with orthogonality yields the orthonormality condition: $\int \psi_m^* \psi_n\,d\tau = \delta_{mn} = \begin{cases} 1 & \text{if } m = n \\ 0 & \text{if } m \ne n \end{cases}$

Commutator Algebra and Compatible Observables

The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as: $[\hat{A}, \hat{B}] \equiv \hat{A}\hat{B} - \hat{B}\hat{A}$

• If $[\hat{A}, \hat{B}] = 0$: The operators commute. The observables can be measured simultaneously with infinite precision, and share a common set of simultaneous eigenfunctions.
• If $[\hat{A}, \hat{B}] \ne 0$: The operators do not commute. The observables cannot be known simultaneously with arbitrary precision, obeying generalized uncertainty $\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|$.

Fundamental Commutator Evaluation: $[\hat{x}, \hat{p}_x] f(x) = x\left(-i\hbar \frac{df}{dx}\right) - \left(-i\hbar \frac{d}{dx}(x f)\right) = -i\hbar x \frac{df}{dx} + i\hbar\left(f + x\frac{df}{dx}\right) = i\hbar f(x)$ $[\hat{x}, \hat{p}_x] = i\hbar \ne 0$ This non-zero commutator is the mathematical origin of the Heisenberg Uncertainty Principle.

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4. The Particle in a One-Dimensional Box

The particle in a 1D box (infinite potential well) is the simplest quantum model exhibiting spatial quantization of energy. Consider a particle of mass $m$ confined along the $x$-axis between impenetrable rigid boundaries: $V(x) = \begin{cases} 0 & 0 < x < L \\ \infty & x \le 0 \text{ and } x \ge L \end{cases}$

Derivation of Energy and Wave Functions

Inside the box ($0 < x < L$), $V(x) = 0$, so the TISE is: $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E\psi \implies \frac{d^2\psi}{dx^2} + k^2 \psi = 0 \quad \text{where } k = \frac{\sqrt{2mE}}{\hbar}$ General solution: $\psi(x) = C \sin(kx) + D \cos(kx)$.

Applying Boundary Conditions (particle cannot penetrate infinite potential, so $\psi = 0$ at boundaries):

1. At $x = 0$: $\psi(0) = C \sin(0) + D \cos(0) = 0 \implies D = 0$.
2. At $x = L$: $\psi(L) = C \sin(kL) = 0$. Because $C \ne 0$ (otherwise $\psi = 0$ everywhere), $\sin(kL) = 0 \implies kL = n\pi$, where $n = 1, 2, 3, \dots$

Quantized energy values follow directly by equating $k$ expressions: $k = \frac{n\pi}{L} = \frac{\sqrt{2mE}}{\hbar} \implies \frac{n^2 \pi^2}{L^2} = \frac{2mE}{\hbar^2} = \frac{2mE}{(h/2\pi)^2} = \frac{8\pi^2 mE}{h^2}$

$E_n = \frac{n^2 h^2}{8 m L^2} \quad (n = 1, 2, 3, \dots)$

Normalizing $\int_0^L |\psi_n(x)|^2 dx = 1$ yields the normalized wave functions:

$\psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n\pi x}{L} \right)$

Vital Physical Deductions

1. Quantum Number $n$: $n$ must be a non-zero positive integer ($1, 2, 3, \dots$). The value $n = 0$ is physically forbidden because $\psi_0(x) = 0$ everywhere, meaning no particle exists.
2. Zero-Point Energy (ZPE): The lowest possible energy state ($n = 1$) has non-zero energy: $E_1 = \frac{h^2}{8 m L^2} > 0$ A confined quantum particle can never be completely motionless. If $E = 0$, linear momentum would be zero with zero uncertainty ($\Delta p = 0$), forcing position uncertainty $\Delta x \to \infty$, which violates confinement in box length $L$.
3. Nodes: An interior node is a point inside the box where $\psi_n(x) = 0$ (excluding walls). A state with quantum number $n$ possesses exactly $n - 1$ interior nodes.
4. Energy Level Spacing: Energy increases with $n^2$, meaning energy level spacing widens as $n$ increases: $\Delta E_{n \to n+1} = E_{n+1} - E_n = (2n + 1) \frac{h^2}{8mL^2}$.

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5. Particle in a Three-Dimensional Box and Degeneracy

For a particle confined in a 3D rectangular box of side lengths $L_x, L_y, L_z$:

$E_{n_x, n_y, n_z} = \frac{h^2}{8m} \left( \frac{n_x^2}{L_x^2} + \frac{n_y^2}{L_y^2} + \frac{n_z^2}{L_z^2} \right) \quad (n_x, n_y, n_z = 1, 2, 3, \dots)$

For a cubic box where $L_x = L_y = L_z = L$: $E = \frac{h^2}{8 m L^2} (n_x^2 + n_y^2 + n_z^2)$

Concept of Degeneracy ($g$)

Degeneracy refers to multiple distinct quantum states (different sets of quantum numbers) that share the exact same energy eigenvalue.

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6. The Quantum Harmonic Oscillator

The harmonic oscillator models molecular vibrations (such as bond stretching in diatomic molecules). The classical restoring force obeys Hooke's Law: $F = -k_{force} x$, corresponding to a parabolic potential energy well:

$V(x) = \frac{1}{2} k_{force} x^2$

Schrödinger Equation and Eigenvalues

$-\frac{\hbar^2}{2\mu} \frac{d^2\psi}{dx^2} + \frac{1}{2} k_{force} x^2 \psi = E\psi$ where $\mu$ is the reduced mass of the vibrating diatomic system.

The allowed quantized vibrational energy levels are:

$E_v = \left( v + \frac{1}{2} \right) h \nu = \left( v + \frac{1}{2} \right) \hbar \omega \quad (v = 0, 1, 2, 3, \dots)$ where $v$ is the vibrational quantum number, and the fundamental classical frequency is: $\nu = \frac{1}{2\pi} \sqrt{\frac{k_{force}}{\mu}}$

Fundamental Properties of the Quantum Harmonic Oscillator

1. Zero-Point Vibrational Energy (ZPVE): In the ground vibrational state ($v = 0$): $E_0 = \frac{1}{2} h \nu = \frac{1}{2} \hbar \omega > 0$ Molecules never cease vibrating, even at absolute zero ($0\text{ K}$).
2. Equidistant Energy Spacing: Unlike the particle in a box (whose spacing widens with $n$), all adjacent harmonic oscillator levels are separated by the exact same constant quantum of energy: $\Delta E = E_{v+1} - E_v = h \nu$
3. Hermite Wave Functions: Wave functions are formulated as Hermite polynomials ($H_v$) modulated by a Gaussian decay envelope: $\psi_v(x) = N_v H_v(\alpha^{1/2} x) e^{-\alpha x^2 / 2} \quad \text{where } \alpha = \frac{\sqrt{\mu k_{force}}}{\hbar}$
   • Ground state ($v = 0$): $H_0 = 1$, wave function is a pure Gaussian curve centered at $x = 0$ with zero nodes.
   • State $v$ possesses exactly $v$ nodes.
4. Quantum Tunneling: In classical physics, an oscillating particle can never venture beyond the "classical turning points" where $E = V(x)$. In quantum mechanics, the wave function decays exponentially into the classically forbidden region, yielding a finite probability of finding the particle where potential energy exceeds total energy ($V(x) > E$).

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7. The Rigid Rotor and Rotational Quantization

A diatomic molecule rotating in three dimensions with fixed bond length $r_0$ and moment of inertia $I = \mu r_0^2$ is modeled as a rigid rotor:

$E_J = \frac{\hbar^2}{2I} J(J + 1) = B J(J + 1) \quad (J = 0, 1, 2, \dots)$ where $J$ is the rotational quantum number, and $B$ is the rotational constant: $B = \frac{\hbar^2}{2I} = \frac{h^2}{8\pi^2 I} \quad \text{(in Joules)}$ In spectroscopic wavenumber units ($\text{cm}^{-1}$): $\tilde{B} = \frac{h}{8\pi^2 c I}$

• Spatial Degeneracy ($g_J$): Each rotational energy level $J$ possesses a $(2J + 1)$-fold spatial degeneracy ($g_J = 2J + 1$), arising from the allowed magnetic projection quantum numbers $M_J \in \{-J, -J+1, \dots, +J\}$.
• Rotational Transition Spacings: Under electric dipole selection rules ($\Delta J = \pm 1$): $\Delta E_{J \to J+1} = E_{J+1} - E_J = B(J+1)(J+2) - BJ(J+1) = 2B(J+1)$ Adjacent absorption lines in pure rotational spectra appear at frequencies $\tilde{\nu} = 2\tilde{B}, 4\tilde{B}, 6\tilde{B}, 8\tilde{B}, \dots$, separated by a constant spacing of $2\tilde{B}$.