Quantum Mechanics - Physics, Philosophy, Mathematics, and Proofs

Introduction

Classical mechanics assumes that a particle has, at every instant, a definite position \(x(t)\) and momentum \(p(t)\). Quantum mechanics replaces this with a state \(|\psi\rangle\) that encodes probability amplitudes for all possible outcomes. Evolution is deterministic via the Schrodinger equation, but predictions for measurements are probabilistic via the Born rule \(P = |\langle a|\psi\rangle|^{2}\).

The theory is not a patch. It is a coherent mathematical structure built on linear vector spaces, unitary evolution, and Hermitian observables. Its success spans chemistry, condensed matter, optics, and particle physics to better than one part in a trillion.

Section 1: Where Did They Get the Notion From?

The quantum hypothesis was forced by data between 1900 and 1925. Classical physics predicted nonsense when applied to light and atoms.

Philosophical roots

Three classical assumptions failed:

1. Continuity of energy: Blackbody radiation and atomic spectra showed discrete lines, not smooth continua.
2. Classical fields for light: Photoelectric and Compton effects required particle-like momentum and energy quanta \(E=h\nu\), \(p=h/\lambda\).
3. Deterministic orbits: Atoms are stable, yet an orbiting classical electron should radiate and collapse in \(10^{-11}\) s. Stability demanded quantized stationary states.

Quantum mechanics came from spectroscopy, calorimetry, and scattering, not from philosophy. The mathematics of Hilbert space was adopted because it naturally encodes superposition and interference observed in experiments.

Section 2: Wave-Particle Duality - Not Paradoxical, but Contextual

Experts do not claim electrons are sometimes waves and sometimes particles. They claim electrons are quantum systems described by a state vector \(|\psi\rangle\). When you measure position, you sample from \(|\psi(x)|^{2}\). When you measure momentum, you sample from \(|\tilde\psi(p)|^{2}\). Both distributions come from the same \(|\psi\rangle\).

Double-slit proof

Let amplitude through slit 1 be \(\psi_{1}\), through slit 2 be \(\psi_{2}\). With both open and no which-path information, total amplitude is

$$ \psi = \psi_{1} + \psi_{2} $$

Probability density:

$$ |\psi|^{2} = |\psi_{1}|^{2} + |\psi_{2}|^{2} + 2\,\text{Re}(\psi_{1}^{*}\psi_{2}) $$

The last term is interference. If a which-path detector entangles with the particle, the states become \(|\psi_{1}\rangle|D_{1}\rangle\) and \(|\psi_{2}\rangle|D_{2}\rangle\) with \(\langle D_{1}|D_{2}\rangle \approx 0\). Then the cross term vanishes after tracing over the detector. This is complementarity, formalized by Bohr in 1927.

Context determines the statistics. The object itself does not switch ontologies.

Section 3: Mathematics of Quantum Mechanics

Postulates

1. State: System state is a normalized vector \(|\psi\rangle\) in a complex Hilbert space \(\mathcal{H}\), \(\langle\psi|\psi\rangle=1\).
2. Observables: Physical quantities correspond to Hermitian operators \(\hat A = \hat A^{\dagger}\) with spectral decomposition \(\hat A = \sum a |a\rangle\langle a|\).
3. Evolution: Closed systems evolve unitarily: \(i\hbar \partial_{t}|\psi\rangle = \hat H |\psi\rangle\). Solution \(|\psi(t)\rangle = \hat U(t)|\psi(0)\rangle\), \(\hat U^{\dagger}\hat U = I\).
4. Born rule: Probability for outcome \(a\) is \(P(a)=|\langle a|\psi\rangle|^{2}\). Expectation \(\langle \hat A\rangle = \langle\psi|\hat A|\psi\rangle\).
5. Projection: After obtaining \(a\), state updates to \(|a\rangle\) (for ideal projective measurement). Modern view treats this as entanglement and decoherence.

Derivation of time-independent Schrodinger equation from de Broglie

Start with plane wave for free particle: \(\psi(x,t)=A e^{i(kx-\omega t)}\). de Broglie relations \(p=\hbar k\), \(E=\hbar\omega\). Then

$$ -i\hbar\partial_{x}\psi = \hbar k \psi = p\psi,\quad -\hbar^{2}\partial_{x}^{2}\psi = p^{2}\psi $$

Classical energy \(E = p^{2}/2m + V(x)\). Replace \(E\to i\hbar\partial_{t}\), \(p^{2}\to -\hbar^{2}\partial_{x}^{2}\):

$$ i\hbar\partial_{t}\psi = -\frac{\hbar^{2}}{2m}\partial_{x}^{2}\psi + V(x)\psi $$

Stationary states \(\psi(x,t)=\phi(x)e^{-iEt/\hbar}\) give

$$ -\frac{\hbar^{2}}{2m}\nabla^{2}\phi + V\phi = E\phi $$

Commutators

On wavefunctions, \(\hat x\psi = x\psi\), \(\hat p\psi = -i\hbar\partial_{x}\psi\). Then

$$ [\hat x,\hat p]\psi = x(-i\hbar\psi') - (-i\hbar)(x\psi)' = i\hbar\psi $$

Thus \([\hat x,\hat p]=i\hbar\hat I\).

Theorem: If \([\hat A,\hat B]\neq 0\), no complete basis of simultaneous eigenstates exists.

Proof: Suppose \(\hat A|a,b\rangle = a|a,b\rangle\) and \(\hat B|a,b\rangle = b|a,b\rangle\). Then \([\hat A,\hat B]|a,b\rangle = (ab-ba)|a,b\rangle =0\), contradicting non-zero commutator. Hence position and momentum cannot both be sharp.

Section 4: Heisenberg Uncertainty Principle - Proof and Meaning

Your question: "how do we know about activity, location, of an atom despite claim that we can from reference of another atom's place."

We can measure the position of atom A relative to your detector, or relative to atom B, with high precision. The uncertainty principle does not forbid that. It states that for any single quantum system, the standard deviations of position and momentum along the same axis satisfy \(\Delta x\,\Delta p \ge \hbar/2\). Measuring \(x_{A}\) sharply increases \(\Delta p_{A}\). The relative coordinate \(x_{\text{rel}}=x_{A}-x_{B}\) also obeys \([x_{\text{rel}}, p_{\text{rel}}]=i\hbar\). Therefore you can know where A is relative to B at an instant, but you cannot simultaneously know their relative momentum precisely. You can locate atoms in a lattice with STM, but you cannot predict their future trajectories with classical certainty because the momentum spread is large.

General derivation

Define \(\Delta \hat A = \hat A - \langle\hat A\rangle\), variance \((\Delta A)^{2} = \langle\psi|(\Delta\hat A)^{2}|\psi\rangle = \| \Delta\hat A|\psi\rangle\|^{2}\). Let \(|f\rangle=\Delta\hat A|\psi\rangle\), \(|g\rangle=\Delta\hat B|\psi\rangle\). By Cauchy-Schwarz:

$$ |\langle f|g\rangle|^{2} \le \langle f|f\rangle\langle g|g\rangle = (\Delta A)^{2}(\Delta B)^{2} $$

Write \(\langle f|g\rangle = \tfrac12\langle\{\Delta\hat A,\Delta\hat B\}\rangle + \tfrac12\langle[\Delta\hat A,\Delta\hat B]\rangle\). The first term is real, second is pure imaginary. Thus

$$ |\langle f|g\rangle|^{2} \ge \tfrac14 |\langle[\hat A,\hat B]\rangle|^{2} $$

Hence

$$ \Delta A\,\Delta B \ge \frac12 |\langle[\hat A,\hat B]\rangle| $$

For \(\hat A = \hat x\), \(\hat B = \hat p\), \([\hat x,\hat p]=i\hbar\):

$$ \Delta x\,\Delta p \ge \hbar/2 $$

Interpretations

Heisenberg's microscope argued measurement disturbs the system. That is true but incomplete. The modern view: \(\Delta x\) and \(\Delta p\) are intrinsic spreads of the wavefunction. A Gaussian wave packet saturates the bound.

Section 5: Philosophy - Interpretations

Bell's theorem (1964) proved no local hidden-variable theory can reproduce all quantum correlations. Experiments since Aspect 1982 violate Bell inequalities, ruling out local realism. The measurement problem remains: all interpretations reproduce the same Born statistics, they differ on ontology, not predictions.

Section 6: Does String Theory Derive From This?

No. String theory does not follow logically from quantum mechanics alone. It is a research program that uses quantum mechanics as a foundation and attempts to include gravity.

Key points:

• String theory replaces point particles with one-dimensional strings. Quantization of string vibrational modes yields a discrete spectrum of particles, including a massless spin-2 mode identified with the graviton.
• It assumes the postulates of QM: states live in Hilbert space, observables are operators, evolution is unitary, \([x,p]=i\hbar\). Without quantization, strings would be classical and unstable.
• Quantum zero-point energy and the uncertainty principle prevent a string from collapsing to a point. The critical dimension (D=26 for bosonic strings, D=10 for superstrings) arises from requiring cancellation of conformal anomaly in the quantum worldsheet theory.
• Therefore string theory is an extension that is consistent with QM and general relativity in certain limits, not a theorem derived from QM. If quantum mechanics were modified, string theory would need reconstruction.

Think of QM as the operating system. String theory is an application that runs on it, aiming to unify forces. It builds on, not derives from, the uncertainty principle and superposition.

Section 7: GNU Octave Examples

Copy these into Octave to verify core results. Units use \(\hbar=1\), \(m=1\) unless noted.

1) infinite_well.m - particle in a box

% infinite_well.m
% Solve -1/2 psi'' = E psi on [0,L] with psi(0)=psi(L)=0
L = 1; N = 1000; x = linspace(0,L,N);
for n = 1:3
  psi = sqrt(2/L)*sin(n*pi*x/L);
  E = (n*pi/L)^2 / 2;
  norm_check = trapz(x, abs(psi).^2); % should be 1
  figure(1); hold on;
  plot(x, psi + n, 'LineWidth', 2); % offset for visibility
  printf('n=%d, E=%.4f, norm=%.6f\n', n, E, norm_check);
end
xlabel('x'); ylabel('\psi_n(x) offset'); title('Infinite well eigenfunctions'); grid on;

2) harmonic_oscillator.m - Hermite solutions

% harmonic_oscillator.m
% V = 1/2 m w^2 x^2, energies En = (n+1/2) hw
hbar = 1; m = 1; w = 1;
x = linspace(-5,5,1000);
for n = 0:3
  Hn = hermite(n, x*sqrt(m*w/hbar)); % Octave polynomial
  norm = 1/sqrt(2^n * factorial(n)) * (m*w/(pi*hbar))^(1/4);
  psi = norm * exp(-m*w*x.^2/(2*hbar)) .* Hn;
  En = (n+0.5)*hbar*w;
  figure(2); hold on; plot(x, psi + n*1.5);
  printf('n=%d, En=%.3f\n', n, En);
end
title('HO eigenfunctions'); grid on;

3) uncertainty_gaussian.m - saturates HUP

% uncertainty_gaussian.m
hbar = 1; sigma = 0.6; % position width
x = linspace(-5,5,2000); dx = x(2)-x(1);
psi_x = (1/(2*pi*sigma^2))^(1/4) * exp(-x.^2/(4*sigma^2));
psi_x = psi_x / sqrt(trapz(x, abs(psi_x).^2));
dx2 = trapz(x, x.^2 .* abs(psi_x).^2); Dx = sqrt(dx2);

k = linspace(-10,10,2000); dk = k(2)-k(1);
psi_k = fftshift(fft(psi_x))*dx/sqrt(2*pi); % FT
psi_k = psi_k / sqrt(trapz(k, abs(psi_k).^2));
p = hbar*k; dp2 = trapz(k, (hbar*k).^2 .* abs(psi_k).^2); Dp = sqrt(dp2);

printf('Δx=%.4f, Δp=%.4f, product=%.4f >= %.4f\n', Dx, Dp, Dx*Dp, hbar/2);

4) tunneling.m - rectangular barrier transmission

% tunneling.m
hbar=1; m=1; V0=1; a=1; % barrier height and width
E = linspace(0.1,2,400);
k1 = sqrt(2*m*E)/hbar;
k2 = sqrt(2*m*(E-V0))/hbar;
% for E < V0, k2 = i*kappa
T = zeros(size(E));
for i=1:length(E)
  if E(i) < V0
    kappa = sqrt(2*m*(V0-E(i)))/hbar;
    T(i) = 1 ./ (1 + (V0^2 * sinh(kappa*a)^2)/(4*E(i)*(V0-E(i))));
  else
    T(i) = 1 ./ (1 + (V0^2 * sin(k2(i)*a)^2)/(4*E(i)*(E(i)-V0)));
  end
end
plot(E, T, 'LineWidth',2); xlabel('E'); ylabel('T'); title('Tunneling transmission'); grid on;

5) double_slit.m - interference

% double_slit.m
lambda=0.05; d=0.5; L=5; % slit separation, screen distance
x = linspace(-2,2,2000);
k = 2*pi/lambda;
r1 = sqrt(L^2 + (x - d/2).^2); r2 = sqrt(L^2 + (x + d/2).^2);
psi1 = exp(1i*k*r1)./sqrt(r1); psi2 = exp(1i*k*r2)./sqrt(r2);
% which-path distinguishability D
for D = [0, 0.5, 0.9]
  V = sqrt(1-D^2); % visibility
  I = abs(psi1).^2 + abs(psi2).^2 + 2*V*real(psi1.*conj(psi2));
  plot(x, I/max(I)); hold on;
end
legend('D=0','D=0.5','D=0.9'); xlabel('screen x'); ylabel('norm intensity'); grid on;

6) hydrogen_radial.m - 1s,2s,2p radial probability

% hydrogen_radial.m
a0=1; r=linspace(0,20,2000);
R1s = 2*exp(-r/a0)/a0^(3/2);
R2s = (1/(2*sqrt(2)*a0^(3/2))).*(2 - r/a0).*exp(-r/(2*a0));
R2p = (1/(2*sqrt(6)*a0^(3/2))).*(r/a0).*exp(-r/(2*a0));
P1s = r.^2 .* abs(R1s).^2; P2s = r.^2 .* abs(R2s).^2; P2p = r.^2 .* abs(R2p).^2;
plot(r,P1s,'LineWidth',2); hold on; plot(r,P2s,'LineWidth',2); plot(r,P2p,'LineWidth',2);
legend('1s','2s','2p'); xlabel('r/a0'); ylabel('r^2|R|^2'); title('Hydrogen radial probability'); grid on;

Section 8: Interactive Quantum Lab

Section 9: Proofs Collection

1. Conservation of normalization

From \(i\hbar\partial_{t}\psi = \hat H\psi\), with \(\hat H^{\dagger}=\hat H\):

$$ \frac{d}{dt}\int |\psi|^{2}dx = \int (\psi^{*}\partial_{t}\psi + \psi\partial_{t}\psi^{*})dx = \frac{1}{i\hbar}\int (\psi^{*}\hat H\psi - (\hat H\psi)^{*}\psi)dx =0 $$

by Hermiticity. Probability is conserved.

2. Ehrenfest theorem

$$ \frac{d}{dt}\langle \hat x\rangle = \frac{1}{i\hbar}\langle[\hat x,\hat H]\rangle = \frac{\langle\hat p\rangle}{m} $$

$$ \frac{d}{dt}\langle \hat p\rangle = \frac{1}{i\hbar}\langle[\hat p, V(\hat x)]\rangle = -\langle \partial_{x}V\rangle $$

Expectation values follow Newton-like equations, but with quantum averages.

3. No simultaneous eigenstates for non-commuting observables

As shown in Section 3, \([\hat A,\hat B]\neq0\) implies no basis \(|a,b\rangle\) with \(\hat A|a,b\rangle=a|a,b\rangle\) and \(\hat B|a,b\rangle=b|a,b\rangle\). Hence \(\Delta A=0\) implies \(\Delta B>0\).

4. No-cloning theorem (sketch)

Suppose unitary \(\hat U\) clones: \(\hat U|\psi\rangle|0\rangle = |\psi\rangle|\psi\rangle\). For two states \(|\psi\rangle,|\phi\rangle\): inner product preservation gives \(\langle\phi|\psi\rangle = (\langle\phi|\psi\rangle)^{2}\). This holds only if \(\langle\phi|\psi\rangle=0\) or \(1\). Arbitrary unknown states cannot be cloned. This protects the uncertainty principle: cloning would allow simultaneous measurement of non-commuting observables.