Fundamental Quantum Mechanics Equations

Why This Matters

Quantum mechanics isn't just a collection of strange formulas—it's a complete mathematical framework that describes reality at its most fundamental level. In this course, you're being tested on your ability to derive, manipulate, and interpret these equations, not just recognize them. The concepts here—wave function evolution, operator algebra, quantization conditions, and measurement theory—form the backbone of everything from atomic physics to quantum computing.

The equations in this guide connect through a beautiful logical structure: the Schrödinger equation tells you how states evolve, operators extract physical information from those states, and commutator relations determine what you can and cannot measure simultaneously. Don't just memorize the formulas—understand what mathematical principle each equation embodies and how they relate to one another. That's what separates a passing score from mastery.

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Wave Function Dynamics and the Schrödinger Equation

The Schrödinger equation is to quantum mechanics what Newton's second law is to classical mechanics—it's the equation of motion that governs how quantum states change.

Schrödinger Equation

• Fundamental equation of quantum mechanics—describes the time evolution of a quantum state through $i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi$
• First-order in time, linear in the wave function—this linearity enables the superposition principle and guarantees unitary (probability-preserving) evolution
• Solutions yield wave functions $\Psi(x,t)$ that encode all measurable information about the system through their complex amplitudes

Time-Independent Schrödinger Equation

• Eigenvalue equation $\hat{H}\psi = E\psi$ used when the potential $V(x)$ has no explicit time dependence
• Determines stationary states and energy eigenvalues—the allowed energies emerge as eigenvalues of the Hamiltonian operator
• Separation of variables yields $\Psi(x,t) = \psi(x)e^{-iEt/\hbar}$, showing that stationary states have time-independent probability densities

Wave Function Normalization

• Physical requirement that total probability equals one: $\int_{-\infty}^{\infty}|\psi(x)|^2\,dx = 1$
• Constrains acceptable solutions—wave functions must be square-integrable (belong to Hilbert space) to represent physical states
• Preserved under time evolution by the Schrödinger equation, ensuring probability conservation

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Operators and Physical Observables

In quantum mechanics, every measurable quantity corresponds to a Hermitian operator. The mathematics of operators—their eigenvalues, eigenfunctions, and commutation relations—determines what we can know about a system.

Hamiltonian Operator

• Total energy operator $\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$ in position representation
• Generator of time evolution—appears in the Schrödinger equation as the operator that "drives" quantum dynamics
• Eigenvalues are allowed energies of the system; finding these is often the central problem in quantum mechanics

Momentum Operator

• Defined in position space as $\hat{p} = -i\hbar\frac{d}{dx}$, where the factor of $-i\hbar$ ensures Hermiticity
• Eigenfunctions are plane waves $e^{ipx/\hbar}$ representing states of definite momentum
• Generates spatial translations—this deep connection between symmetries and conservation laws underlies Noether's theorem in quantum form

Position Operator

• Acts by multiplication in position representation: $\hat{x}\psi(x) = x\psi(x)$
• Eigenfunctions are delta functions $\delta(x - x_0)$, representing perfectly localized (but unphysical) states
• Non-commuting with momentum—this fundamental incompatibility leads directly to the uncertainty principle

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Measurement Theory and Probability

Quantum mechanics is fundamentally probabilistic. These equations tell you how to extract predictions from wave functions and what limits exist on simultaneous knowledge.

Born Rule (Probability Density)

• Probability density given by $P(x) = |\psi(x)|^2 = \psi^*(x)\psi(x)$, the squared modulus of the wave function
• Connects mathematics to experiment—this interpretive rule transforms abstract wave functions into measurable predictions
• Probability of finding particle in region $[a,b]$ is $\int_a^b |\psi(x)|^2\,dx$; this is what detectors actually measure

Expectation Value

• Average measurement outcome calculated as $\langle \hat{A} \rangle = \int_{-\infty}^{\infty} \psi^*(x)\hat{A}\psi(x)\,dx$
• Bracket notation $\langle \psi | \hat{A} | \psi \rangle$ emphasizes the inner product structure of quantum mechanics
• Not necessarily an eigenvalue—the expectation value can take any value in the spectrum's range, representing statistical averaging

Heisenberg Uncertainty Principle

• Fundamental measurement limit $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\Delta$ denotes standard deviation
• Derived from commutator relations—for any operators, $\Delta A \Delta B \geq \frac{1}{2}|\langle[\hat{A},\hat{B}]\rangle|$
• Not about measurement disturbance—reflects intrinsic quantum indeterminacy; conjugate variables cannot simultaneously have sharp values

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Commutator Algebra and Operator Structure

Commutators reveal the deep structure of quantum mechanics. Whether two observables can be simultaneously measured, and how operators transform under symmetries, all follow from commutation relations.

Commutator Relations

• Definition $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$ measures the failure of operators to commute
• Canonical commutator $[\hat{x}, \hat{p}] = i\hbar$ is the foundation of quantum mechanics—all uncertainty relations trace back to this
• Zero commutator implies compatible observables—if $[\hat{A}, \hat{B}] = 0$, the operators share eigenstates and can be simultaneously measured

Angular Momentum Operators

• Components $\hat{L}_x, \hat{L}_y, \hat{L}_z$ satisfy cyclic commutation relations $[\hat{L}_x, \hat{L}_y] = i\hbar\hat{L}_z$ (and cyclic permutations)
• Total angular momentum $\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$ commutes with all components: $[\hat{L}^2, \hat{L}_i] = 0$
• Eigenvalues are quantized: $L^2 = \hbar^2 l(l+1)$ and $L_z = \hbar m$ with $m = -l, ..., +l$

Spin Operators (Pauli Matrices)

• Intrinsic angular momentum for spin-1/2 particles: $\hat{S}_i = \frac{\hbar}{2}\sigma_i$ where $\sigma_i$ are Pauli matrices
• Pauli matrices $\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$, $\sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}$, $\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
• Fundamental to quantum information—spin-1/2 systems (qubits) form the basis of quantum computing

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Exactly Solvable Systems

These canonical problems aren't just textbook exercises—they're the building blocks for understanding real physical systems and the testing ground for quantum mechanical techniques.

Particle in a Box (Infinite Potential Well)

• Boundary conditions $\psi(0) = \psi(L) = 0$ force quantization: only standing waves with nodes at walls are allowed
• Energy eigenvalues $E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}$ with $n = 1, 2, 3, ...$ (note: ground state is $n=1$, not zero)
• Wave functions $\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right)$ form a complete orthonormal basis

Harmonic Oscillator

• Energy levels $E_n = \hbar\omega\left(n + \frac{1}{2}\right)$ are evenly spaced with zero-point energy $\frac{1}{2}\hbar\omega$
• Ladder operators $\hat{a}^\dagger$ and $\hat{a}$ raise and lower energy eigenstates, enabling elegant algebraic solutions
• Ubiquitous model—describes molecular vibrations, phonons, photon modes, and any system near a potential minimum

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Relativistic Quantum Mechanics

When particles move at speeds comparable to light, quantum mechanics must merge with special relativity. The Dirac equation accomplishes this for fermions.

Dirac Equation

• Relativistic wave equation $(i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0$ that's first-order in both space and time derivatives
• Predicts antimatter—negative energy solutions correspond to positrons; this was confirmed experimentally four years after Dirac's prediction
• Spin emerges naturally—unlike non-relativistic quantum mechanics, spin-1/2 isn't added by hand but follows from requiring Lorentz invariance

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Quick Reference Table

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Self-Check Questions

1. The canonical commutator $[\hat{x}, \hat{p}] = i\hbar$ directly implies which fundamental principle? Write out the general uncertainty relation it produces.

2. Compare and contrast the energy spectra of the particle in a box and harmonic oscillator. Which system has a zero-point energy, and why does the other not have equally spaced levels?

3. Given a wave function $\psi(x)$, explain the mathematical steps to calculate (a) the probability of finding the particle between $x = 0$ and $x = a$, and (b) the expectation value of momentum.

4. Why can we simultaneously know $\hat{L}^2$ and $\hat{L}_z$ for a quantum system, but not $\hat{L}_x$ and $\hat{L}_z$? What mathematical property determines this?

5. An FRQ presents a time-independent potential and asks for the allowed energies. Outline the general procedure: what equation do you solve, what boundary/normalization conditions apply, and how do eigenvalues emerge?