Quantum Mechanics
Heisenberg 1925 · Schrödinger 1926 · Dirac 1930 · Born Rule · Bell 1964
I. Historical Origins & the Quantum Revolution
Classical physics failed catastrophically at atomic scales. Three experiments forced a radical rethinking of nature's laws:
| Year | Problem | Classical Failure | Quantum Resolution |
|---|---|---|---|
| 1900 | Blackbody radiation | Ultraviolet catastrophe — \(\infty\) energy predicted | Planck: \(E = n\hbar\omega\), energy quantized |
| 1905 | Photoelectric effect | Intensity, not frequency, should eject electrons | Einstein: photons with \(E = \hbar\omega\) |
| 1913 | Atomic spectra | No stable orbits; spiraling collapse | Bohr: quantized angular momentum \(L = n\hbar\) |
| 1922 | Stern-Gerlach | Continuous deflection expected | Discrete spin states \(m_s = \pm\tfrac{1}{2}\) |
| 1923 | Compton scattering | Thomson scattering: no frequency shift | Photon momentum \(p = \hbar k\) |
| 1924 | de Broglie hypothesis | — | All matter has wavelength \(\lambda = h/p\) |
| 1925 | Heisenberg matrix mechanics | — | Observables are non-commuting matrices |
| 1926 | Schrödinger equation | — | Wave equation for matter; identical to Heisenberg's |
The de Broglie relations unify particle and wave descriptions:
$$E = \hbar\omega = hf,\qquad \mathbf{p} = \hbar\mathbf{k},\qquad \lambda = \frac{h}{p}$$II. The Six Postulates of Quantum Mechanics
The complete state of a quantum system is encoded in a normalized vector \(\ket{\psi}\) (or wavefunction \(\psi(\mathbf{x},t)\)) in a complex Hilbert space \(\mathcal{H}\). All physical predictions are derived from \(\ket{\psi}\).
$$\braket{\psi}{\psi} = 1$$Every physical observable \(A\) corresponds to a Hermitian (self-adjoint) operator \(\op{A}\) on \(\mathcal{H}\), satisfying \(\op{A}^\dagger = \op{A}\). Hermiticity guarantees real eigenvalues (physical measurement outcomes).
Measurement of \(\op{A}\) yields eigenvalue \(a_n\) with probability \(P(a_n) = |\braket{a_n}{\psi}|^2\), where \(\ket{a_n}\) is the eigenvector. Immediately after, the state collapses to \(\ket{a_n}\).
$$\ev{\op{A}} = \braket{\psi}{\op{A}|\psi} = \sum_n a_n |\braket{a_n}{\psi}|^2$$Between measurements, the state evolves unitarily according to:
$$\boxed{i\hbar\frac{\partial}{\partial t}\ket{\psi(t)} = \op{H}\ket{\psi(t)}}$$where \(\op{H}\) is the Hamiltonian operator (total energy).
Classical canonical variables \((q_i, p_j)\) become operators with commutation relations:
$$\comm{\op{q}_i}{\op{p}_j} = i\hbar\delta_{ij},\qquad \comm{\op{q}_i}{\op{q}_j} = \comm{\op{p}_i}{\op{p}_j} = 0$$In the position representation: \(\op{x} = x\,\cdot\) and \(\op{p} = -i\hbar\nabla\).
The wavefunction of a system of identical particles must be symmetric (bosons) or antisymmetric (fermions) under exchange of any two particles.
III. Dirac (Bra-Ket) Notation
Kets, Bras, and Inner Products
| Symbol | Name | Meaning |
|---|---|---|
| \(\ket{\psi}\) | Ket | State vector in \(\mathcal{H}\) |
| \(\bra{\psi}\) | Bra | Dual vector (conjugate transpose) |
| \(\braket{\phi}{\psi}\) | Bracket | Inner product: \(\int\phi^*\psi\,d^3x\) |
| \(\ket{\psi}\bra{\phi}\) | Outer product | Operator (rank-1 projection) |
| \(\braket{\psi|\op{A}|\psi}\) | Matrix element | Expectation value of \(\op{A}\) |
Completeness (Resolution of Identity)
For any orthonormal basis \(\{\ket{n}\}\):
$$\sum_n \ket{n}\bra{n} = \op{I} \quad\text{(discrete)},\qquad \int\ket{x}\bra{x}\,dx = \op{I}\quad\text{(continuous)}$$Position and Momentum Representations
$$\psi(x) = \braket{x}{\psi},\qquad \tilde\psi(p) = \braket{p}{\psi} = \frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^\infty \psi(x)\,e^{-ipx/\hbar}\,dx$$These are Fourier transform pairs. The momentum eigenstate in position space is:
$$\braket{x}{p} = \frac{1}{\sqrt{2\pi\hbar}}\,e^{ipx/\hbar}$$Hermitian Operators
\(\op{A}\) is Hermitian if \(\op{A}^\dagger = \op{A}\), i.e., \(\braket{\phi|\op{A}|\psi} = \braket{\psi|\op{A}|\phi}^*\) for all \(\ket{\phi},\ket{\psi}\in\mathcal{H}\).
Consequences: (1) Real eigenvalues. (2) Eigenvectors with different eigenvalues are orthogonal. (3) Eigenvectors form a complete basis.
Let \(\op{A}\ket{a} = a\ket{a}\). Then \(\braket{a|\op{A}|a} = a\braket{a}{a} = a\). But also \(\braket{a|\op{A}|a} = \braket{a|\op{A}^\dagger|a}^* = \braket{a|\op{A}|a}^* = a^*\). Hence \(a = a^*\), so \(a\in\mathbb{R}\).
IV. Hilbert Spaces & Linear Algebra
A Hilbert space \(\mathcal{H}\) is a complete inner product space over \(\mathbb{C}\). For quantum mechanics: \(L^2(\mathbb{R}^3)\) — square-integrable functions with \(\langle f,g\rangle = \int f^*g\,d^3x\).
Spectral Theorem
Every Hermitian operator \(\op{A}\) has a spectral decomposition:
$$\op{A} = \sum_n a_n\ket{a_n}\bra{a_n}\quad\text{or}\quad \op{A} = \int a\,\ket{a}\bra{a}\,da$$Functions of operators: \(f(\op{A}) = \sum_n f(a_n)\ket{a_n}\bra{a_n}\).
Unitary Evolution
The time-evolution operator \(\op{U}(t) = e^{-i\op{H}t/\hbar}\) is unitary (\(\op{U}^\dagger\op{U}=\op{I}\)), preserving norms (probability). Formally:
$$\ket{\psi(t)} = e^{-i\op{H}t/\hbar}\ket{\psi(0)} = \op{U}(t)\ket{\psi(0)}$$Change of Basis & Representations
$$A_{mn} = \braket{m|\op{A}|n}\quad\text{(matrix elements)},\qquad \op{A}\ket{n} = \sum_m A_{mn}\ket{m}$$Two representations of quantum mechanics: Schrödinger picture (states evolve, operators fixed) and Heisenberg picture (operators evolve, states fixed).
$$\frac{d\op{A}_H}{dt} = \frac{i}{\hbar}\comm{\op{H}}{\op{A}_H} + \left(\frac{\partial\op{A}}{\partial t}\right)_H$$V. The Schrödinger Equation — Derivation & Meaning
Step 1. A free particle with definite momentum \(p\) and energy \(E\) has de Broglie wavelength \(\lambda = h/p\), represented by plane wave:
$$\psi(x,t) = A\exp\!\left(\frac{i(px - Et)}{\hbar}\right)$$Step 2. Compute derivatives:
$$\frac{\partial\psi}{\partial t} = -\frac{iE}{\hbar}\psi \implies i\hbar\frac{\partial\psi}{\partial t} = E\psi$$ $$\frac{\partial^2\psi}{\partial x^2} = -\frac{p^2}{\hbar^2}\psi \implies -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} = \frac{p^2}{2m}\psi = T\psi$$Step 3. Classical energy \(E = T + V = p^2/(2m) + V(x)\) then gives:
$$\boxed{i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi = \op{H}\psi}$$In 3D
$$i\hbar\frac{\partial\Psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right]\Psi = \op{H}\Psi$$Properties of the Schrödinger Equation
- Linear: if \(\psi_1,\psi_2\) are solutions, so is \(c_1\psi_1+c_2\psi_2\) (superposition principle)
- First-order in time: initial state \(\psi(x,0)\) determines all future evolution
- Norm-preserving: \(d/dt\int|\psi|^2\,dx = 0\) (unitarity)
- Complex: necessarily complex — real wave equation has real solutions; \(\hbar\) requires complex \(\psi\)
VI. The Born Rule & Probability Current
The probability density of finding the particle at position \(x\) at time \(t\) is \(|\psi|^2\). Normalization requires \(\int_{-\infty}^\infty|\psi|^2\,dx = 1\).
Continuity Equation & Probability Current
Compute \(\partial_t|\psi|^2 = \psi^*\partial_t\psi + \psi\partial_t\psi^*\). From TDSE: \(\partial_t\psi = (i\hbar/2m)\nabla^2\psi - (i/\hbar)V\psi\) and its conjugate. Substituting:
$$\partial_t|\psi|^2 = \frac{i\hbar}{2m}(\psi^*\nabla^2\psi - \psi\nabla^2\psi^*) = \frac{i\hbar}{2m}\nabla\cdot(\psi^*\nabla\psi - \psi\nabla\psi^*)$$Identifying \(\mathbf{J} = \frac{\hbar}{2mi}(\psi^*\nabla\psi - \psi\nabla\psi^*)\) gives \(\partial_t\rho + \nabla\cdot\mathbf{J} = 0\). ✓
VII. Stationary States & the TISE
For a time-independent Hamiltonian, separate \(\Psi(x,t) = \psi(x)\cdot T(t)\):
$$i\hbar\frac{T'}{T} = \frac{1}{\psi}\left(-\frac{\hbar^2}{2m}\psi''+V\psi\right) = E = \text{const}$$Time dependence: \(\quad\Psi_n(x,t) = \psi_n(x)\,e^{-iE_n t/\hbar}\)
Properties of Stationary States
- They are energy eigenstates: \(\op{H}\psi_n = E_n\psi_n\)
- Probability density \(|\Psi_n|^2 = |\psi_n|^2\) is time-independent
- Every observable expectation is constant in time: \(d\ev{\op{A}}/dt = 0\) if \([\op{A},\op{H}]=0\)
- General solution: \(\Psi(x,t) = \sum_n c_n\psi_n(x)e^{-iE_nt/\hbar}\), with \(\sum_n|c_n|^2=1\)
VIII. Operators, Eigenvalues & Expectation Values
Key Operators in Quantum Mechanics
| Observable | Operator (position space) | Eigenvalue equation |
|---|---|---|
| Position \(x\) | \(\op{x} = x\cdot\) | \(x\,\delta(x-x_0) = x_0\,\delta(x-x_0)\) |
| Momentum \(p\) | \(\op{p} = -i\hbar\partial_x\) | \(-i\hbar\partial_x\,e^{ipx/\hbar} = p\,e^{ipx/\hbar}\) |
| Kinetic energy | \(\op{T} = -\hbar^2\nabla^2/2m\) | Plane waves \(e^{i\mathbf{k}\cdot\mathbf{r}}\) |
| Hamiltonian | \(\op{H} = \op{T}+V(\op{x})\) | \(\op{H}\psi_n = E_n\psi_n\) |
| Angular momentum \(L_z\) | \(-i\hbar\partial_\phi\) | \(e^{im\phi}\), \(m\in\mathbb{Z}\) |
| Parity | \(\op{\Pi}:\psi(x)\to\psi(-x)\) | \(\pm 1\) |
Variance & Standard Deviation
$$\sigma_A^2 = \ev{(\op{A}-\ev{\op{A}})^2} = \ev{\op{A}^2} - \ev{\op{A}}^2$$IX. Commutators & Compatible Observables
If \(\comm{\op{A}}{\op{B}} = 0\): compatible — both can be simultaneously measured with definite values; they share a common eigenbasis.
If \(\comm{\op{A}}{\op{B}} \ne 0\): incompatible — cannot both have definite values (uncertainty principle).
Fundamental Commutation Relations
$$\comm{\op{x}}{\op{p}} = i\hbar,\qquad\comm{\op{x}^n}{\op{p}} = i\hbar n\op{x}^{n-1},\qquad\comm{\op{x}}{f(\op{p})} = i\hbar f'(\op{p})$$ $$\comm{\op{L}_x}{\op{L}_y} = i\hbar\op{L}_z,\quad\comm{\op{L}_y}{\op{L}_z} = i\hbar\op{L}_x,\quad\comm{\op{L}_z}{\op{L}_x} = i\hbar\op{L}_y$$ $$\comm{\op{L}^2}{\op{L}_i} = 0\text{ for all }i \quad\Rightarrow\quad L^2\text{ and }L_z\text{ simultaneously measurable}$$Acting on test function \(\phi(x)\):
$$(\op{x}\op{p} - \op{p}\op{x})\phi = x\left(-i\hbar\frac{d\phi}{dx}\right) - \left(-i\hbar\frac{d}{dx}\right)(x\phi)$$ $$= -i\hbar x\phi' + i\hbar(\phi + x\phi') = i\hbar\phi$$Since this holds for all \(\phi\): \(\comm{\op{x}}{\op{p}} = i\hbar\op{I}\).
X. Free Particle & Wave Packets
With \(V=0\), the TISE gives plane wave solutions \(\psi_k(x)=Ae^{ikx}\) with \(E=\hbar^2k^2/(2m)\). These are not normalizable alone — the physical states are wave packets:
$$\Psi(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \phi(k)\,e^{i(kx-\omega(k)t)}\,dk,\qquad \omega(k) = \frac{\hbar k^2}{2m}$$Gaussian Wave Packet
At \(t=0\), centered at \(x_0\) with mean momentum \(p_0 = \hbar k_0\) and width \(\sigma\):
$$\psi(x,0) = \left(\frac{1}{2\pi\sigma^2}\right)^{1/4}\exp\!\left(-\frac{(x-x_0)^2}{4\sigma^2}\right)\exp(ik_0 x)$$At time \(t\), the packet spreads:
$$|\Psi(x,t)|^2 = \frac{1}{\sqrt{2\pi}\,w(t)}\exp\!\left(-\frac{(x-x_0-v_g t)^2}{2w(t)^2}\right)$$ $$w(t) = \sigma\sqrt{1+\left(\frac{\hbar t}{2m\sigma^2}\right)^2},\qquad v_g = \frac{\hbar k_0}{m} = \frac{p_0}{m}$$XI. The Infinite Square Well (Particle in a Box)
\(V(x) = 0\) for \(0\le x\le L\), \(V=\infty\) outside. Boundary conditions: \(\psi(0)=\psi(L)=0\).
Inside the well: TISE gives \(\psi'' = -k^2\psi\) with \(k=\sqrt{2mE}/\hbar\). General solution: \(\psi = A\sin(kx)+B\cos(kx)\).
BC at \(x=0\): \(\psi(0)=0 \Rightarrow B=0\).
BC at \(x=L\): \(\psi(L)=A\sin(kL)=0 \Rightarrow kL = n\pi\), \(n=1,2,3,\ldots\)
Energy quantization:
$$E_n = \frac{\hbar^2\pi^2 n^2}{2mL^2} = \frac{n^2\pi^2\hbar^2}{2mL^2},\quad n=1,2,3,\ldots$$Normalization: \(\int_0^L A^2\sin^2(n\pi x/L)\,dx = A^2 L/2 = 1 \Rightarrow A = \sqrt{2/L}\).
$$\boxed{\psi_n(x) = \sqrt{\frac{2}{L}}\sin\!\left(\frac{n\pi x}{L}\right),\quad E_n = \frac{n^2\pi^2\hbar^2}{2mL^2}}$$Properties
- Zero-point energy: \(E_1 = \pi^2\hbar^2/(2mL^2) > 0\) — particle can never be at rest (uncertainty principle)
- Orthonormality: \(\int_0^L\psi_m^*\psi_n\,dx = \delta_{mn}\)
- Parity: odd \(n\) → symmetric about center; even \(n\) → antisymmetric
- Nodes: \(\psi_n\) has \(n-1\) interior nodes
- Spacing between levels: \(E_{n+1}-E_n = (2n+1)E_1\) (non-uniform → non-classical)
Expectation Values in State \(\psi_n\)
$$\ev{\op{x}} = \frac{L}{2},\quad \ev{\op{p}} = 0,\quad \ev{\op{x}^2} = \frac{L^2}{3}\left(1-\frac{3}{2n^2\pi^2}\right),\quad \ev{\op{p}^2} = \frac{n^2\pi^2\hbar^2}{L^2}$$ $$\sigma_x\sigma_p = \frac{n\pi\hbar}{\sqrt{3}}\sqrt{\frac{1}{3}-\frac{1}{2n^2\pi^2}} \ge \frac{\hbar}{2}\quad\checkmark$$XII. The Finite Square Well
\(V(x) = -V_0\) for \(|x|\le a\), \(V=0\) outside. Bound states (\(-V_0 < E < 0\)) have exponential decay into classically forbidden regions.
Transcendental Equations for Bound States
Let \(\kappa = \sqrt{-2mE}/\hbar\), \(l = \sqrt{2m(E+V_0)}/\hbar\). Matching conditions give:
$$\text{Even: }\kappa = l\tan(la),\qquad \text{Odd: }\kappa = -l\cot(la)$$These are solved graphically or numerically. There is always at least one bound state for any \(V_0 > 0\), but only finitely many. Wide/deep wells → many states approaching the infinite-well limit.
Tunneling (Evanescent Waves)
In the classically forbidden region \(E < V\), the wavefunction doesn't vanish but decays exponentially:
$$\psi \propto e^{-\kappa x},\quad \kappa = \frac{\sqrt{2m(V-E)}}{\hbar}$$This quantum tunneling is responsible for alpha decay, scanning tunneling microscopy (STM), and tunnel diodes.
XIII. The Quantum Harmonic Oscillator
The most important exactly solvable system: \(V = \frac{1}{2}m\omega^2 x^2\), \(\op{H} = \frac{\op{p}^2}{2m} + \frac{1}{2}m\omega^2\op{x}^2\).
Ladder Operator Method
Define dimensionless operators:
$$\op{a} = \sqrt{\frac{m\omega}{2\hbar}}\left(\op{x}+\frac{i\op{p}}{m\omega}\right),\quad \op{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(\op{x}-\frac{i\op{p}}{m\omega}\right)$$Step 1 — Commutator:
$$\comm{\op{a}}{\op{a}^\dagger} = \frac{m\omega}{2\hbar}\left[\op{x}+\frac{i\op{p}}{m\omega},\,\op{x}-\frac{i\op{p}}{m\omega}\right] = \frac{m\omega}{2\hbar}\left(-\frac{2i}{m\omega}\comm{\op{x}}{\op{p}}\right) = \frac{m\omega}{2\hbar}\cdot\frac{2\hbar}{m\omega} = 1$$Step 2 — Rewrite Hamiltonian:
$$\op{a}^\dagger\op{a} = \frac{m\omega}{2\hbar}\left(\op{x}^2+\frac{\op{p}^2}{m^2\omega^2}+\frac{i}{m\omega}\comm{\op{x}}{\op{p}}\right) = \frac{\op{H}}{\hbar\omega}-\frac{1}{2}$$ $$\boxed{\op{H} = \hbar\omega\left(\op{a}^\dagger\op{a}+\half\right) = \hbar\omega\left(\op{N}+\half\right),\quad \op{N} = \op{a}^\dagger\op{a}}$$Step 3 — Ladder structure: From \(\comm{\op{N}}{\op{a}^\dagger} = \op{a}^\dagger\) and \(\comm{\op{N}}{\op{a}} = -\op{a}\):
$$\op{N}\op{a}^\dagger\ket{n} = (n+1)\op{a}^\dagger\ket{n},\quad \op{N}\op{a}\ket{n} = (n-1)\op{a}\ket{n}$$So \(\op{a}^\dagger\ket{n} \propto\ket{n+1}\) (raises energy) and \(\op{a}\ket{n} \propto\ket{n-1}\) (lowers energy).
Step 4 — Ground state: Since \(n\ge 0\) (energy \(\ge 0\)), there must exist \(\ket{0}\) with \(\op{a}\ket{0}=0\). Normalized:
$$\op{a}^\dagger\ket{n} = \sqrt{n+1}\ket{n+1},\quad\op{a}\ket{n} = \sqrt{n}\ket{n-1}$$ $$\boxed{E_n = \hbar\omega\left(n+\half\right),\quad n=0,1,2,3,\ldots}$$Wavefunctions
Ground state from \(\op{a}\psi_0 = 0\): \(\psi_0(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-m\omega x^2/(2\hbar)}\)
Excited states via \(\psi_n = (a^\dagger)^n\psi_0/\sqrt{n!}\):
$$\psi_n(x) = \frac{1}{\sqrt{2^n n!}}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}H_n\!\left(\sqrt{\frac{m\omega}{\hbar}}\,x\right)e^{-m\omega x^2/(2\hbar)}$$where \(H_n\) are Hermite polynomials: \(H_0=1,\;H_1=2\xi,\;H_2=4\xi^2-2,\;H_3=8\xi^3-12\xi,\ldots\)
Zero-Point Energy and Uncertainty
Ground state: \(E_0 = \hbar\omega/2 > 0\). This is required by the uncertainty principle — a particle cannot be at rest at the potential minimum.
$$\sigma_x = \sqrt{\frac{\hbar}{2m\omega}},\quad\sigma_p = \sqrt{\frac{m\omega\hbar}{2}},\quad\sigma_x\sigma_p = \frac{\hbar}{2}\quad\text{(minimum uncertainty!)}$$XIV. Delta Function Potential
\(V(x) = -\alpha\delta(x)\), \(\alpha > 0\). Only one bound state:
$$\psi(x) = \frac{\sqrt{m\alpha}}{\hbar}\,e^{-m\alpha|x|/\hbar^2},\qquad E = -\frac{m\alpha^2}{2\hbar^2}$$Matching conditions at \(x=0\): \(\psi\) is continuous; \(\psi'\) has a finite jump \(\Delta\psi' = -2m\alpha\psi(0)/\hbar^2\) (from integrating TISE across \(x=0\)).
Scattering off a Delta Barrier (\(\alpha < 0\))
$$T = \frac{1}{1 + m\alpha^2/(2\hbar^2 E)},\qquad R = \frac{m\alpha^2/(2\hbar^2 E)}{1+m\alpha^2/(2\hbar^2 E)},\quad R+T=1$$XV. The Hydrogen Atom
Central potential \(V(r) = -e^2/(4\pi\varepsilon_0 r)\). In 3D with spherical symmetry, separate \(\psi(r,\theta,\phi) = R_{nl}(r)\,Y_l^m(\theta,\phi)\).
Spherical Harmonics
Angular eigenstates of \(\op{L}^2\) and \(\op{L}_z\):
$$\op{L}^2 Y_l^m = \hbar^2 l(l+1)Y_l^m,\qquad\op{L}_z Y_l^m = \hbar m Y_l^m$$ $$Y_0^0 = \frac{1}{\sqrt{4\pi}},\quad Y_1^0 = \sqrt{\frac{3}{4\pi}}\cos\theta,\quad Y_1^{\pm1} = \mp\sqrt{\frac{3}{8\pi}}\sin\theta\,e^{\pm i\phi}$$Energy Levels & Quantum Numbers
The quantum numbers and their ranges:
| Symbol | Name | Range | Physical meaning |
|---|---|---|---|
| \(n\) | Principal | \(1,2,3,\ldots\) | Energy: \(E_n \propto -1/n^2\) |
| \(l\) | Orbital | \(0,1,\ldots,n-1\) | Orbital angular momentum \(\sqrt{l(l+1)}\hbar\) |
| \(m\) | Magnetic | \(-l,\ldots,+l\) | z-component \(L_z = m\hbar\) |
| \(m_s\) | Spin | \(\pm\frac{1}{2}\) | Spin z-component \(S_z = m_s\hbar\) |
Radial Wavefunctions
$$R_{nl}(r) = -\sqrt{\left(\frac{2}{na_0}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}}\,e^{-r/(na_0)}\left(\frac{2r}{na_0}\right)^l L_{n-l-1}^{2l+1}\!\left(\frac{2r}{na_0}\right)$$where \(a_0 = 4\pi\varepsilon_0\hbar^2/(m_e e^2) \approx 0.529\,\)Å is the Bohr radius, and \(L\) are associated Laguerre polynomials. First few:
$$R_{10} = \frac{2}{a_0^{3/2}}e^{-r/a_0},\quad R_{20} = \frac{1}{\sqrt{2}a_0^{3/2}}\left(1-\frac{r}{2a_0}\right)e^{-r/(2a_0)},\quad R_{21} = \frac{r}{2\sqrt{6}a_0^{5/2}}e^{-r/(2a_0)}$$Degeneracy
Each energy \(E_n\) has degeneracy \(n^2\) (counting \(l\) from \(0\) to \(n-1\), and \(m\) from \(-l\) to \(l\)). With spin: \(2n^2\) states per shell.
XVI. The Uncertainty Principle — Robertson Theorem
For position and momentum: \(\sigma_x\sigma_p \ge \hbar/2\).
Define \(\ket{f} = (\op{A}-\ev{A})\ket{\psi}\) and \(\ket{g} = (\op{B}-\ev{B})\ket{\psi}\). Then \(\sigma_A^2 = \braket{f}{f}\) and \(\sigma_B^2 = \braket{g}{g}\).
By the Cauchy-Schwarz inequality: \(\braket{f}{f}\braket{g}{g} \ge |\braket{f}{g}|^2\).
Write \(\braket{f}{g} = \tfrac{1}{2}\braket{f|g+g|f} + \tfrac{1}{2}\braket{f|g-g|f}\). The first term is real (call it \(X\)); the second is \(\tfrac{1}{2}\braket{\psi|\comm{\op{A}}{\op{B}}|\psi} = \tfrac{i}{2}\ev{[\op{A},\op{B}]}\) (imaginary).
Therefore: \(|\braket{f}{g}|^2 = X^2 + \tfrac{1}{4}|\ev{\comm{\op{A}}{\op{B}}}|^2 \ge \tfrac{1}{4}|\ev{\comm{\op{A}}{\op{B}}}|^2\).
Taking square roots: \(\sigma_A\sigma_B \ge \tfrac{1}{2}|\ev{\comm{\op{A}}{\op{B}}}|\).
Minimum Uncertainty States
The inequality is saturated (\(\sigma_x\sigma_p = \hbar/2\)) if and only if \(\ket{g} = i\lambda\ket{f}\) (proportional) and \(X=0\). For position-momentum, this gives exactly the Gaussian wave packet — coherent states of the harmonic oscillator.
XVII. Ehrenfest's Theorem & Classical Limit
Using \(\partial_t\ket\psi = -\frac{i}{\hbar}\op{H}\ket\psi\) and its conjugate:
$$= \frac{i}{\hbar}\braket{\psi|\op{H}\op{A}-\op{A}\op{H}|\psi}+\left\langle\pd{\op{A}}{t}\right\rangle = \frac{i}{\hbar}\ev{\comm{\op{H}}{\op{A}}}+\left\langle\pd{\op{A}}{t}\right\rangle$$Newton's Laws Recovered
Apply to position and momentum (no explicit time dependence):
$$\frac{d\ev{x}}{dt} = \frac{i}{\hbar}\ev{\comm{\op{H}}{\op{x}}} = \frac{\ev{\op{p}}}{m}\quad\checkmark$$ $$\frac{d\ev{p}}{dt} = \frac{i}{\hbar}\ev{\comm{\op{H}}{\op{p}}} = -\left\langle\frac{dV}{dx}\right\rangle \approx -\frac{dV}{dx}\bigg|_{x=\ev{x}}$$The last approximation (valid for slowly varying \(V\)) gives \(m\ddot{\ev{x}} = -dV/dx\) — exactly Newton's second law for the mean position. Quantum mechanics reproduces classical mechanics in the appropriate limit.
XVIII. Angular Momentum
The angular momentum operator \(\op{\mathbf{L}} = \op{\mathbf{r}}\times\op{\mathbf{p}}\) with components satisfying \(\comm{L_i}{L_j} = i\hbar\varepsilon_{ijk}L_k\).
Raising/Lowering Operators
$$\op{L}_\pm = \op{L}_x \pm i\op{L}_y,\qquad\comm{\op{L}_z}{\op{L}_\pm} = \pm\hbar\op{L}_\pm,\qquad\comm{\op{L}^2}{\op{L}_\pm}=0$$ $$\op{L}_+\ket{l,m} = \hbar\sqrt{l(l+1)-m(m+1)}\ket{l,m+1}$$ $$\op{L}_-\ket{l,m} = \hbar\sqrt{l(l+1)-m(m-1)}\ket{l,m-1}$$Eigenvalue Quantization — Proof
Since \(\op{L}^2 - \op{L}_z^2 = \op{L}_x^2+\op{L}_y^2 \ge 0\), we have \(\hbar^2[l(l+1)-m^2]\ge 0\), so \(|m|\le\sqrt{l(l+1)}\). The ladder \(\op{L}_+^k\ket{l,m}\) must terminate at some \(m_\text{max}\), giving \(\op{L}_+\ket{l,m_\text{max}}=0 \Rightarrow m_\text{max} = l\). Similarly \(m_\text{min}=-l\). The number of steps \(2l\) must be a non-negative integer, so \(l\in\{0,\tfrac{1}{2},1,\tfrac{3}{2},\ldots\}\).
XIX. Spin & Pauli Matrices
Spin is an intrinsic quantum property with no classical analog. For spin-\(\tfrac{1}{2}\) particles (electrons, quarks, protons):
$$\op{S}_i = \frac{\hbar}{2}\sigma_i,\qquad \sigma_x = \begin{pmatrix}0&1\\1&0\end{pmatrix},\quad\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$$Properties of Pauli Matrices
$$\sigma_i\sigma_j = \delta_{ij}I + i\varepsilon_{ijk}\sigma_k,\qquad\sigma_i^2 = I,\qquad\mathrm{Tr}(\sigma_i)=0,\qquad\det(\sigma_i)=-1$$ $$\{\sigma_i,\sigma_j\} = 2\delta_{ij}I\quad\text{(anticommutator)},\qquad [\sigma_i,\sigma_j] = 2i\varepsilon_{ijk}\sigma_k$$Spinors and Spin States
$$\ket{+z} = \chi_+ = \begin{pmatrix}1\\0\end{pmatrix},\quad\ket{-z} = \chi_- = \begin{pmatrix}0\\1\end{pmatrix}$$ $$\ket{+x} = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix},\quad\ket{+y}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\i\end{pmatrix}$$Spin Precession in Magnetic Field
Hamiltonian: \(\op{H} = -\boldsymbol{\mu}\cdot\mathbf{B} = \frac{g_s e}{2m_e}\op{\mathbf{S}}\cdot\mathbf{B}\). For \(\mathbf{B}=B\hat{z}\): spin precesses at Larmor frequency \(\omega_L = g_s eB/(2m_e)\).
XX. Identical Particles & Quantum Statistics
Pauli Exclusion Principle
For fermions, the antisymmetric wavefunction must be written as a Slater determinant:
$$\psi(1,\ldots,N) = \frac{1}{\sqrt{N!}}\begin{vmatrix}\psi_a(1)&\psi_a(2)&\cdots\\\psi_b(1)&\psi_b(2)&\cdots\\\vdots&&\ddots\end{vmatrix}$$If any two fermions are in the same state (\(\psi_a=\psi_b\)), two rows are identical → determinant vanishes → forbidden. This is the Pauli exclusion principle, responsible for the periodic table, white dwarf stability, and neutron stars.
Exchange Interaction and Fermi Pressure
Even without any interaction potential, identical fermions experience an effective repulsion (exchange energy) due to antisymmetrization. Bosons experience effective attraction (Bose-Einstein condensation).
XXI. Entanglement & Bell's Theorem
Entangled States
A two-particle state is entangled if it cannot be written as a product \(\ket{\psi_1}\otimes\ket{\psi_2}\). The canonical example is the Bell state (EPR pair):
$$\ket{\Phi^+} = \frac{1}{\sqrt{2}}\left(\ket{\uparrow\uparrow}+\ket{\downarrow\downarrow}\right)$$Measuring particle 1 in state \(\ket{\uparrow}\) instantaneously collapses particle 2 to \(\ket{\uparrow}\), regardless of separation. This is not a signal — the outcome is random and cannot be used for faster-than-light communication (no-communication theorem).
Bell's Theorem (1964)
Any local hidden variable (LHV) theory must satisfy Bell inequalities. For the CHSH form, with measurement settings \(a,a',b,b'\):
$$|\ev{AB}+\ev{AB'}+\ev{A'B}-\ev{A'B'}| \le 2\quad\text{(classical, LHV)}$$Quantum mechanics predicts (and experiments confirm): maximum value \(2\sqrt{2} \approx 2.83\). This is a violation by \(\sqrt{2}\) — quantum correlations are stronger than any local hidden variable theory can produce. Experiments by Aspect (1982), Zeilinger (2015), and others definitively ruled out LHV theories.
XXII. Perturbation Theory
Solve \(\op{H} = \op{H}^{(0)} + \lambda\op{H}'\) when \(\op{H}^{(0)}\) is exactly solvable and \(\op{H}'\) is small.
First-Order Non-Degenerate Perturbation Theory
Second-Order Energy Correction
$$E_n^{(2)} = \sum_{m\ne n}\frac{|\braket{m^{(0)}|\op{H}'|n^{(0)}}|^2}{E_n^{(0)}-E_m^{(0)}}$$Note: \(E_n^{(2)} < 0\) for the ground state (the denominator is always negative) — perturbations always lower the ground state energy at second order.
Time-Dependent Perturbation Theory & Fermi's Golden Rule
For a periodic perturbation \(\op{H}'=V\cos(\omega t)\), the transition rate from state \(\ket{i}\) to \(\ket{f}\) is:
$$\Gamma_{i\to f} = \frac{2\pi}{\hbar}|\braket{f|V|i}|^2\,\rho(E_f)\quad\text{(Fermi's Golden Rule)}$$where \(\rho(E_f)\) is the density of final states. This governs spontaneous emission, radioactive decay, and scattering cross-sections.
XXIII. WKB (Semiclassical) Approximation
Valid when the de Broglie wavelength \(\lambda = 2\pi\hbar/p(x)\) varies slowly. Expand \(\psi = e^{iS(x)/\hbar}\):
$$\psi(x) \approx \frac{C}{\sqrt{p(x)}}\exp\!\left(\pm\frac{i}{\hbar}\int p(x)\,dx\right),\quad p(x) = \sqrt{2m(E-V(x))}$$WKB Quantization (Bohr-Sommerfeld)
$$\oint p(x)\,dx = 2\pi\hbar\left(n+\frac{1}{2}\right),\quad n=0,1,2,\ldots$$Tunneling Amplitude
Through a barrier from \(x_1\) to \(x_2\) where \(E < V(x)\):
$$T \approx \exp\!\left(-\frac{2}{\hbar}\int_{x_1}^{x_2}\sqrt{2m(V(x)-E)}\,dx\right)$$Applied to alpha decay (Gamow): \(\Gamma \propto e^{-2G}\), where \(G\) is the Gamow factor. Explains the enormous range of alpha-decay half-lives (femtoseconds to billion years).
XXIV. The Variational Method
For any normalized trial state \(\ket{\psi_\text{trial}}\):
$$E_\text{gs} \le \ev{\op{H}}_\text{trial} = \braket{\psi_\text{trial}|\op{H}|\psi_\text{trial}}$$Equality holds iff \(\ket{\psi_\text{trial}}\) is the exact ground state.
Expand in energy eigenbasis: \(\ket{\psi_\text{trial}} = \sum_n c_n\ket{n}\). Then:
$$\ev{H}_\text{trial} = \sum_n |c_n|^2 E_n \ge E_\text{gs}\sum_n|c_n|^2 = E_\text{gs}$$Strategy: choose a trial wavefunction with adjustable parameter(s) \(\alpha\), minimize \(\ev{H}(\alpha)\) over \(\alpha\). Used in: helium ground state, molecular bonding, density functional theory (DFT).
XXV. Feynman Path Integrals
Feynman (1948) reformulated QM: the propagator \(K(x_f,t_f;x_i,t_i)\) is a sum over all possible paths:
$$K(x_f,x_i;T) = \int\mathcal{D}[x(t)]\,\exp\!\left(\frac{i}{\hbar}\int_0^T L(x,\dot x)\,dt\right)$$where \(L = T - V\) is the classical Lagrangian. Each path contributes with amplitude \(e^{iS/\hbar}\) where \(S\) is the classical action.
Classical Limit
As \(\hbar\to 0\), the phase \(e^{iS/\hbar}\) oscillates wildly for all paths except near the stationary point \(\delta S = 0\) — the classical path (stationary phase approximation). This recovers the principle of least action.
Connection to Statistical Mechanics
Under Wick rotation \(t\to -i\tau\) (imaginary time), the path integral becomes the partition function \(Z = \int\mathcal{D}[x]\,e^{-S_E/\hbar}\), connecting quantum mechanics to classical statistical mechanics at temperature \(T = \hbar/k_B\tau\).
XXVI. Toward Quantum Field Theory
QM breaks down at relativistic energies and for particle creation/annihilation. Quantum Field Theory (QFT) promotes fields to operators:
Second Quantization
$$\op{\psi}(\mathbf{r}) = \sum_k \op{a}_k u_k(\mathbf{r}),\qquad\comm{\op{a}_k}{\op{a}_{k'}^\dagger} = \delta_{kk'}\text{ (bosons)},\quad\acomm{\op{a}_k}{\op{a}_{k'}^\dagger}=\delta_{kk'}\text{ (fermions)}$$Particle number is no longer fixed. The Fock space vacuum \(\ket{0}\) has zero particles; states \(\ket{n_1,n_2,\ldots}\) specify occupation numbers.
Standard Model Interactions
| Force | Gauge boson | Coupling | QFT |
|---|---|---|---|
| Electromagnetic | Photon \(\gamma\) | \(\alpha \approx 1/137\) | QED (Feynman, Schwinger, Tomonaga 1948) |
| Weak | \(W^\pm, Z^0\) | \(\alpha_W \approx 0.03\) | Electroweak (Glashow, Weinberg, Salam 1968) |
| Strong | Gluons \(g\) | \(\alpha_s \approx 0.1\) | QCD (Gross, Politzer, Wilczek 1973) |
XXVII. GNU Octave Examples
1 — Numerical Eigenvalues via Finite Difference (TISE Solver)
%% Numerically solve the Time-Independent Schrödinger Equation %% -ℏ²/(2m) ψ'' + V(x)ψ = Eψ using finite difference discretization %% Works for ANY potential V(x) — infinite well, harmonic, double well, etc. clear; clc; %% ─── Physical parameters (atomic units: ℏ=m=1) ────────────── hbar = 1; m = 1; L = 10; % domain [-L/2, L/2] N = 500; % grid points x = linspace(-L/2, L/2, N); dx = x(2)-x(1); %% ─── Define potential (change this for different systems) ───── potential = 'harmonic'; % 'infinite_well', 'harmonic', 'double_well', 'morse' switch potential case 'infinite_well' omega = 1; V = zeros(1,N); % V=0 inside (walls enforced via Dirichlet BC) E_analytic = (1:8).^2 * pi^2*hbar^2/(2*m*L^2); case 'harmonic' omega = 1; V = 0.5 * m * omega^2 * x.^2; E_analytic = hbar*omega*(((0:7))+0.5); % (n+½)ℏω case 'double_well' a = 1.5; b = 0.5; V = a * x.^4 - b * x.^2; % Mexican hat V = V - min(V); E_analytic = []; case 'morse' De = 4; a_m = 0.8; x0 = 0; V = De * (1 - exp(-a_m*(x-x0))).^2; E_analytic = []; end %% ─── Build tridiagonal Hamiltonian matrix ──────────────────── %% Kinetic: -ℏ²/(2m) * (ψ_{i+1}-2ψ_i+ψ_{i-1})/dx² diag_main = hbar^2/(m*dx^2) * ones(1,N) + V; diag_off = -hbar^2/(2*m*dx^2) * ones(1,N-1); H = diag(diag_main) + diag(diag_off,1) + diag(diag_off,-1); %% ─── Solve eigenvalue problem ──────────────────────────────── [vecs, vals] = eig(H); E = diag(vals); [E, idx] = sort(E); vecs = vecs(:,idx); %% Normalise eigenvectors: ∫|ψ|²dx = 1 for k = 1:8 vecs(:,k) = vecs(:,k) / sqrt(trapz(x, vecs(:,k).^2)); end %% ─── Print energy levels ───────────────────────────────────── printf('%-6s %-14s', 'n', 'E_numeric'); if ~isempty(E_analytic), printf('%-14s %-12s','E_analytic','Error'); end printf('\n%s\n', repmat('-',1,46)); for k = 1:8 printf('%-6d %-14.6f', k, E(k)); if ~isempty(E_analytic) printf('%-14.6f %-12.2e', E_analytic(k), abs(E(k)-E_analytic(k))); end printf('\n'); end %% ─── Plot wavefunctions and probability densities ──────────── figure(1); clf; colors = jet(8); subplot(1,2,1); hold on; grid on; title(sprintf('Wavefunctions ψ_n(x) — %s', potential)); xlabel('x (a.u.)'); ylabel('ψ_n(x) + E_n (offset)'); % Plot V(x) plot(x, V, 'k-', 'LineWidth', 2); for k = 1:5 psi_plot = vecs(:,k)' / 2; % scale for display plot(x, psi_plot + E(k), 'Color', colors(k,:), 'LineWidth', 1.5); plot(x([1 end]), [E(k) E(k)], 'Color', colors(k,:), 'LineStyle', '--'); end subplot(1,2,2); hold on; grid on; title('Probability Densities |ψ_n(x)|²'); xlabel('x'); ylabel('|ψ|² + E_n'); plot(x, V, 'k-', 'LineWidth', 2); for k = 1:5 rho = vecs(:,k)'.^2 / 3; fill([x, fliplr(x)], [rho+E(k), E(k)*ones(1,N)], ... colors(k,:), 'FaceAlpha', 0.4, 'EdgeColor', 'none'); end
2 — Wave Packet Time Evolution via FFT
%% Time evolution of a Gaussian wave packet using split-operator FFT method %% Ψ(x,t+dt) ≈ e^{-iVdt/2ℏ} · FFT^{-1}[e^{-iTdt/ℏ} · FFT[e^{-iVdt/2ℏ}Ψ]] %% Highly accurate: 4th order in dt, exact for each operator separately clear; clc; hbar = 1; m = 1; N = 1024; L = 60; x = linspace(-L/2, L/2-L/N, N); dx = L/N; %% Momentum grid (FFT-ordered) dk = 2*pi/L; k = [0:N/2-1, -N/2:-1] * dk; %% Initial Gaussian wave packet: position x0, width sigma, momentum k0 x0 = -10; sigma = 1.5; k0 = 2.5; psi = (2*pi*sigma^2)^(-0.25) * exp(-((x-x0).^2)/(4*sigma^2)) .* exp(1i*k0*x); psi = psi / sqrt(trapz(x, abs(psi).^2)); % normalize %% Potential: Gaussian barrier + square well V_barrier = 3 * exp(-(0.8*x).^2); % barrier at origin V = V_barrier; %% Pre-compute propagators dt = 0.04; expV_half = exp(-1i*V/(hbar) * dt/2); % half-step in V expT_full = exp(-1i*hbar*k.^2/(2*m) * dt); % full step in T %% Animate time evolution figure(2); clf; T_final = 20; nsteps = round(T_final/dt); snap_times = round(linspace(0, nsteps, 6)); snaps = {}; t = 0; for step = 0:nsteps if ismember(step, snap_times) snaps{end+1} = struct('psi', psi, 't', t); end % Split-operator step psi = expV_half .* psi; psi = ifft(expT_full .* fft(psi)); psi = expV_half .* psi; t += dt; end for ii = 1:length(snaps) subplot(2,3,ii); rho = abs(snaps{ii}.psi).^2; plot(x, rho, 'b-', 'LineWidth', 1.5); hold on; plot(x, V_barrier/10, 'r--', 'LineWidth', 1); title(sprintf('t = %.1f ℏ/E', snaps{ii}.t)); xlabel('x'); ylabel('|ψ|²'); grid on; ylim([0 0.5]); end %% Uncertainty principle verification rho_x = abs(snaps{1}.psi).^2; rho_p = abs(fft(snaps{1}.psi)*dx/sqrt(2*pi)).^2; sigma_x = sqrt(trapz(x, x.^2.*rho_x) - trapz(x, x.*rho_x)^2); sigma_p = sigma * ... % for Gaussian: σ_p = ℏ/(2σ_x) printf('σ_x·σ_p = %.4f (should be ≥ %.4f = ℏ/2)\n', sigma_x*hbar/(2*sigma_x), hbar/2);
3 — Harmonic Oscillator: Matrix Method & Hermite Polynomials
%% Harmonic oscillator: matrix representation of ladder operators %% Verify E_n = ℏω(n+½) and compute wavefunctions via Hermite polynomials clear; clc; hbar = 1; m = 1; omega = 1; N_trunc = 20; % Fock space truncation: |0⟩....|N-1⟩ %% ─── Ladder operators in Fock basis ───────────────────────── % a|n⟩ = sqrt(n)|n-1⟩ → a_{m,n} = sqrt(n) δ_{m,n-1} n_arr = 0:N_trunc-1; a = diag(sqrt(1:N_trunc-1), 1); % annihilation ad = a'; % creation N_op = ad*a; % number operator %% Hamiltonian H = ℏω(N + ½) H = hbar*omega*(N_op + 0.5*eye(N_trunc)); %% Verify: eigenvalues should be (n+½)ℏω E_matrix = diag(H); E_exact = hbar*omega*(((0:N_trunc-1))+0.5); printf('Max eigenvalue error: %.2e\n', max(abs(E_matrix - E_exact'))); %% Position and momentum operators xi_scale = sqrt(hbar/(2*m*omega)); x_op = xi_scale * (a + ad); % x̂ = √(ℏ/2mω)(a+a†) p_op = 1i*sqrt(m*omega*hbar/2) * (ad - a); % p̂ %% Verify [x̂,p̂] = iℏ (check matrix element ⟨0|[x,p]|0⟩ = iℏ) comm_xp = x_op*p_op - p_op*x_op; printf('[x,p]_{00}/iℏ = %.6f (should be 1)\n', real(comm_xp(1,1)/(1i*hbar))); %% ─── Hermite polynomial wavefunctions ─────────────────────── function H_n = hermite(n, xi) if n==0, H_n = ones(1,length(xi)); elseif n==1, H_n = 2*xi; else H_prev2 = ones(1,length(xi)); H_prev1 = 2*xi; for k = 2:n H_n = 2*xi.*H_prev1 - 2*(k-1)*H_prev2; H_prev2 = H_prev1; H_prev1 = H_n; end end end xi_max = 5; xi = linspace(-xi_max, xi_max, 500); x_phys = xi * sqrt(hbar/(m*omega)); % dimensionful position figure(3); clf; subplot(1,2,1); hold on; grid on; title('Harmonic Oscillator ψ_n(x)'); V_plot = 0.5 * x_phys.^2; plot(x_phys, V_plot, 'k-', 'LineWidth', 1.5); c2 = jet(6); for n = 0:5 norm_n = 1/sqrt(2^n * factorial(n)) * (m*omega/(pi*hbar))^(0.25); psi_n = norm_n * hermite(n, xi) .* exp(-xi.^2/2); E_n = hbar*omega*(n+0.5); plot(x_phys, psi_n*0.6+E_n, 'Color', c2(n+1,:), 'LineWidth', 1.5); plot(x_phys([1,end]),[E_n E_n],'Color',c2(n+1,:),'LineStyle','--','LineWidth',0.8); text(xi_max*0.8*sqrt(hbar/(m*omega)), E_n+0.15, sprintf('n=%d',n), 'Color', c2(n+1,:)); end xlabel('x'); ylabel('E_n + ψ_n (scaled)'); %% ─── Time-evolved coherent state ───────────────────────────── % Coherent state |α⟩ with α=2: minimum-uncertainty state alpha = 2; c_n = exp(-abs(alpha)^2/2) * alpha.^(0:N_trunc-1) ./ sqrt(factorial(0:N_trunc-1)); printf('Coherent state norm: %.6f\n', sum(abs(c_n).^2)); t_arr = linspace(0, 2*pi/omega, 6); subplot(1,2,2); hold on; grid on; title('Coherent State |α=2⟩ Evolution'); for it = 1:length(t_arr) tt = t_arr(it); psi_coh = zeros(1,length(xi)); for n = 0:N_trunc-1 norm_n = 1/sqrt(2^n*factorial(n)) * (m*omega/(pi*hbar))^(0.25); psi_n = norm_n * hermite(n, xi) .* exp(-xi.^2/2); psi_coh += c_n(n+1) * exp(-1i*omega*(n+0.5)*tt) * psi_n; end rho = abs(psi_coh).^2; plot(x_phys, rho + it/2, 'Color', hsv2rgb([(it-1)/6, 0.9, 0.9]), 'LineWidth', 1.4); end xlabel('x'); ylabel('|ψ|² + offset');
4 — Hydrogen Atom Radial Wavefunctions
%% Hydrogen atom radial wavefunctions R_{nl}(r) and probability densities %% Using associated Laguerre polynomials L^{2l+1}_{n-l-1}(2r/na₀) clear; clc; a0 = 1; % Bohr radius (atomic units) %% Associated Laguerre polynomial L^k_n(x) via recurrence function L = assoc_laguerre(n, k, x) if n==0, L = ones(size(x)); elseif n==1, L = 1 + k - x; else L0 = ones(size(x)); L1 = 1+k-x; for j = 2:n L2 = ((2*j-1+k-x).*L1 - (j-1+k)*L0) / j; L0 = L1; L1 = L2; end L = L1; end end %% Radial wavefunction R_{nl}(r) function R = radial_wf(n, l, r, a0) rho = 2*r/(n*a0); norm = -sqrt((2/(n*a0))^3 * factorial(n-l-1) / (2*n*factorial(n+l)^3)); R = norm .* exp(-rho/2) .* rho.^l .* assoc_laguerre(n-l-1, 2*l+1, rho); end %% Plot for n=1,2,3 and all l r = linspace(0, 30, 1000); states = {[1,0],[2,0],[2,1],[3,0],[3,1],[3,2]}; labs = {'1s','2s','2p','3s','3p','3d'}; figure(4); clf; subplot(1,2,1); hold on; grid on; title('Radial Wavefunctions R_{nl}(r)'); xlabel('r/a₀'); ylabel('R_{nl}(r) a₀^{3/2}'); c3 = lines(6); for k = 1:length(states) n = states{k}(1); l = states{k}(2); R = radial_wf(n, l, r, a0); plot(r, R, 'Color', c3(k,:), 'LineWidth', 1.6, 'DisplayName', labs{k}); end legend('Location','northeast'); ylim([-0.8 1.8]); subplot(1,2,2); hold on; grid on; title('Radial Probability Density r²|R_{nl}|²'); xlabel('r/a₀'); ylabel('r²|R|²'); for k = 1:length(states) n = states{k}(1); l = states{k}(2); R = radial_wf(n, l, r, a0); P = r.^2 .* R.^2; plot(r, P, 'Color', c3(k,:), 'LineWidth', 1.6, 'DisplayName', labs{k}); %% Verify normalization: ∫₀^∞ r²|R|² dr = 1 norm_check = trapz(r, P); printf('%-4s: ∫r²|R|²dr = %.4f\n', labs{k}, norm_check); end legend('Location','northeast'); %% ─── Expectation value ⟨r⟩ = n²a₀(1 + ½(1-l(l+1)/n²)) ──── printf('\nExpectation values ⟨r⟩:\n'); for k = 1:length(states) n=states{k}(1); l=states{k}(2); R = radial_wf(n, l, r, a0); r_mean = trapz(r, r.^3 .* R.^2); r_exact= a0/2 * (3*n^2 - l*(l+1)); printf('%s: ⟨r⟩_num=%.3f ⟨r⟩_exact=%.3f a₀\n', labs{k}, r_mean, r_exact); end
5 — First-Order Perturbation Theory: Anharmonic Oscillator
%% First and second-order perturbation theory for H = H₀ + λx⁴ %% H₀ = harmonic oscillator; H' = x⁴ (quartic correction) %% Compare perturbative to numerical (exact matrix diagonalization) clear; clc; hbar = 1; m = 1; omega = 1; N = 30; % Fock space size lambda = 0.05; % perturbation strength (small) %% Build ladder operators a = diag(sqrt(1:N-1), 1); ad = a'; x_scale = sqrt(hbar/(2*m*omega)); x_op = x_scale * (a + ad); %% H₀ = ℏω(N+½), H' = x⁴ H0 = hbar*omega * (ad*a + 0.5*eye(N)); H1 = x_op^4; % x̂⁴ matrix H_full = H0 + lambda * H1; %% ─── Perturbation theory formulas ─────────────────────────── E0 = hbar*omega * ((0:N-1) + 0.5); % unperturbed energies % First order: E_n^(1) = λ⟨n|x⁴|n⟩ E1_pert = lambda * diag(H1)'; % diagonal matrix elements % Analytical first order: ⟨n|x⁴|n⟩ = (ℏ/2mω)² (6n²+6n+3) x_scale2 = (hbar/(2*m*omega))^2; n_arr = 0:N-1; E1_analytic = lambda * x_scale2 * (6*n_arr.^2 + 6*n_arr + 3); % Second order: sum over m≠n of |⟨m|H'|n⟩|²/(E_n-E_m) E2_pert = zeros(1,10); for n = 1:10 for mm = 1:N if mm ~= n E2_pert(n) += lambda^2 * abs(H1(mm,n))^2 / (E0(n)-E0(mm)); end end end %% Exact (full matrix diagonalization) [~, E_exact] = eig(H_full); E_exact = sort(diag(E_exact)); %% Compare printf('%-4s %-12s %-12s %-12s %-12s %-10s\n', 'n','E₀','E₀+E₁','E₀+E₁+E₂','E_exact','Error₂'); for n = 1:8 E_2nd = E0(n)+E1_pert(n)+E2_pert(n); printf('%-4d %-12.5f %-12.5f %-12.5f %-12.5f %-10.2e\n', n-1, E0(n), E0(n)+E1_pert(n), E_2nd, E_exact(n), abs(E_2nd-E_exact(n))); end %% Plot: perturbative vs exact for varying λ lambdas = linspace(0, 0.5, 50); n_show = 0; % ground state E_pert_arr = zeros(1,length(lambdas)); E_exact_arr= zeros(1,length(lambdas)); for il = 1:length(lambdas) lam = lambdas(il); H_l = H0 + lam * H1; E_exact_arr(il) = min(eig(H_l)); E_pert_arr(il) = E0(1) + lam*E1_analytic(1) + lam^2*E2_pert(1)/lambda^2; end figure(5); plot(lambdas, E_exact_arr,'b-',lambdas,E_pert_arr,'r--','LineWidth',2); legend({'Exact','2nd-order PT'}); grid on; title('Ground State Energy: Exact vs Perturbation Theory'); xlabel('\lambda'); ylabel('E_0(\lambda)');
6 — Quantum Tunneling via Transfer Matrix
%% Transfer matrix method for quantum tunneling through arbitrary barriers %% Exact for piecewise-constant potentials; approximates any smooth barrier clear; clc; hbar = 1; m = 1; %% ─── Transfer matrix for a single slab [x1,x2] with potential V ──── function M = transfer_matrix(E, V, dx, hbar, m) if E > V k = sqrt(2*m*(E-V))/hbar; M = [cos(k*dx), sin(k*dx)/k; -k*sin(k*dx), cos(k*dx)]; else kappa = sqrt(2*m*(V-E))/hbar; M = [cosh(kappa*dx), sinh(kappa*dx)/kappa; kappa*sinh(kappa*dx), cosh(kappa*dx)]; end end %% ─── Rectangular barrier: V₀ for |x| < a ──────────────────── V0 = 5; a = 1; % barrier height and half-width N_slabs = 200; x_bar = linspace(-a, a, N_slabs); dx = 2*a/N_slabs; V_slabs = V0 * ones(1, N_slabs); %% Scan energy E from 0 to 2*V0 E_arr = linspace(0.05, 2*V0, 300); T_num = zeros(1, length(E_arr)); T_wkb = zeros(1, length(E_arr)); for iE = 1:length(E_arr) E = E_arr(iE); M_total = eye(2); for j = 1:N_slabs M_total = transfer_matrix(E, V_slabs(j), dx, hbar, m) * M_total; end % T = 1/|M(2,2)|² (via M matrix extraction) k_out = sqrt(2*m*max(E,0.01))/hbar; T_num(iE) = 1/(1 + (abs(M_total(2,1))/k_out)^2); % WKB formula (for E < V0) if E < V0 kappa_avg = sqrt(2*m*(V0-E))/hbar; T_wkb(iE) = exp(-2*kappa_avg*2*a); else T_wkb(iE) = NaN; end end % Analytic result for rectangular barrier (E < V0): % T = 1/(1 + V₀²sinh²(2κa)/(4E(V₀-E))) T_analytic = zeros(size(E_arr)); for iE = 1:length(E_arr) E = E_arr(iE); if E < V0 kappa = sqrt(2*m*(V0-E))/hbar; T_analytic(iE) = (1 + V0^2*sinh(kappa*2*a)^2/(4*E*(V0-E)))^-1; else k = sqrt(2*m*(E-V0))/hbar; k0 = sqrt(2*m*E)/hbar; T_analytic(iE) = (1 + V0^2*sin(k*2*a)^2/(4*E*(E-V0)))^-1; end end figure(6); semilogy(E_arr, T_num, 'b-', E_arr, T_analytic, 'r--', ... E_arr, T_wkb, 'g:', 'LineWidth', 2); xline(V0, 'k--', 'V₀'); legend({'Transfer Matrix','Analytic','WKB'}); title('Transmission Coefficient vs Energy'); xlabel('E/V₀'); ylabel('T(E)'); grid on; %% Note resonances at E > V₀ where 2ka = nπ (Ramsauer-Townsend effect)
XXVIII. Interactive Simulations
Simulation I — Infinite Square Well: Energy Levels & Wavefunctions
Adjust \(n\) to see \(\psi_n(x)\), \(|\psi_n|^2\), and energy \(E_n = n^2 E_1\). The energy level diagram shows quantization. Toggle superposition of two states to see beating.
Simulation II — Gaussian Wave Packet: Spreading & Uncertainty
Watch the wave packet propagate and spread. The width \(w(t) = \sigma\sqrt{1+(ℏt/2mσ²)²}\) grows — a direct consequence of \(\sigma_x\sigma_p \ge ℏ/2\).
Simulation III — Double-Slit Interference
Young's double slit with a quantum particle. The probability distribution \(|\psi|^2 \propto \cos^2(\pi d\sin\theta/\lambda)\cdot(\sin(\pi a\sin\theta/\lambda)/(\pi a\sin\theta/\lambda))^2\) shows interference × diffraction.
Simulation IV — Harmonic Oscillator Eigenstates & Coherent States
View eigenstates \(\psi_n\) built from Hermite polynomials, or animate the coherent state — a minimum uncertainty wavepacket that oscillates classically without spreading.
Simulation V — Quantum Tunneling Through a Barrier
Shows transmitted and reflected probability amplitudes as a function of energy. The tunneling probability decays exponentially with barrier width and height — the WKB approximation \(T\approx e^{-2\kappa a}\).