Part I — The Local Hidden Variable Hypothesis
Einstein was deeply uncomfortable with quantum mechanics. His objection was not that it gave wrong predictions, but that it seemed incomplete. His view: quantum randomness is like classical statistical mechanics — an artifact of ignorance about underlying deterministic variables. There must be hidden variables $\lambda$ that, if known, would determine all outcomes with certainty.
A local hidden variable (LHV) theory formalizes this idea precisely. Each particle carries a hidden variable $\lambda$ drawn from some distribution $p(\lambda)$. When Alice measures spin along direction $\hat{n}$, the outcome is a deterministic function $A(\hat{n}, \lambda) \in \{+1, -1\}$. Similarly for Bob: $B(\hat{m}, \lambda) \in \{+1, -1\}$. The key locality assumption:
- $A$ does not depend on Bob's detector setting $\hat{m}$
- $B$ does not depend on Alice's detector setting $\hat{n}$
The two-particle correlation in a LHV theory is:
The question Bell asked: is there any choice of $A$, $B$, and $p(\lambda)$ that reproduces all quantum mechanical predictions?
Part II — Bell's Inequality (CHSH)
Bell derived a constraint that any LHV theory must satisfy. The modern form, due to Clauser, Horne, Shimony, and Holt, uses four detector settings: Alice has directions $a, a'$ and Bob has directions $b, b'$. Define correlations $E(a,b) = \langle A(a)B(b)\rangle$. The CHSH inequality is:
The proof is elementary. For fixed $\lambda$, define $A = A(a,\lambda)$, $A' = A(a',\lambda)$, $B = B(b,\lambda)$, $B' = B(b',\lambda)$, each $\in \{+1,-1\}$. Consider:
Since $B, B' \in \{+1,-1\}$, exactly one of $(B-B')$ or $(B+B')$ equals zero and the other equals $\pm 2$. In both cases $|S(\lambda)| = 2$. After averaging over $\lambda$:
Part III — Quantum Violation of Bell's Inequality
For the singlet state, the quantum mechanical correlation between measurements along directions $\hat{n}$ and $\hat{m}$ is:
where $\theta_{nm}$ is the angle between the two directions. Choosing the optimal angles $a = 0°$, $a' = 90°$, $b = 45°$, $b' = 135°$:
| Pair | Angle | Correlation |
|---|---|---|
| $E(a,b)$ | $45°$ | $-\cos 45° = -1/\sqrt{2}$ |
| $E(a,b')$ | $135°$ | $-\cos 135° = +1/\sqrt{2}$ |
| $E(a',b)$ | $45°$ | $-\cos 45° = -1/\sqrt{2}$ |
| $E(a',b')$ | $45°$ | $-\cos 45° = -1/\sqrt{2}$ |
The value $2\sqrt{2}$ is the quantum maximum — the Tsirelson bound. It is achieved by the singlet with this specific angle configuration. Quantum mechanics not only violates the LHV bound; it saturates its own maximum.
Part IV — What Bell's Theorem Actually Proves
Bell's theorem is a mathematical theorem, not an experimental result. The structure of the argument is a logical chain:
- Assume LHV. Then necessarily $|S| \leq 2$ — this is the proven mathematical bound.
- Experiment measures $|S| \approx 2\sqrt{2}$. Therefore LHV is false.
What must be given up? The LHV framework has two key assumptions:
- Realism: outcomes are determined by pre-existing values of $\lambda$.
- Locality: Alice's outcome doesn't depend on Bob's setting, and vice versa.
Bell's theorem says you can't have both. Most interpretations preserve locality (no faster-than-light signals are possible with entangled particles — you cannot use them to communicate) and abandon hidden variable realism. Bohr's response: the question "what is the spin before measurement?" is simply meaningless — there is no spin value before it is measured.
Part V — Continuum States: From Discrete to Continuous
All our quantum mechanics so far has involved finite-dimensional Hilbert spaces — the qubit, multi-spin systems. Now we turn to a particle on a line. The new observable is position $\hat{x}$ with a continuous spectrum $x \in \mathbb{R}$.
The position eigenstates $|x\rangle$ satisfy $\hat{x}|x\rangle = x|x\rangle$ and are labeled by a continuous parameter. The normalization and completeness relations replace their discrete counterparts:
| Property | Discrete | Continuous |
|---|---|---|
| Normalization | $\langle n|m\rangle = \delta_{nm}$ | $\langle x|x'\rangle = \delta(x-x')$ |
| Completeness | $\sum_n |n\rangle\langle n| = 1$ | $\int |x\rangle\langle x|\,dx = 1$ |
The wave function is the projection of the state onto position eigenstates:
The physical interpretation: $|\psi(x)|^2\,dx$ is the probability of finding the particle between $x$ and $x + dx$. The normalization condition becomes $\int_{-\infty}^\infty |\psi(x)|^2\,dx = 1$.
Part VI — The Dirac Delta Function
The Dirac delta $\delta(x - x_0)$ appears everywhere in continuous quantum mechanics. It is not an ordinary function — it is a distribution (generalized function) defined by its action on test functions:
Key properties:
- $\delta(-x) = \delta(x)$ — even function
- $x\,\delta(x) = 0$
- $\delta(ax) = \delta(x)/|a|$ for $a \neq 0$
- Fourier representation: $\delta(x) = \dfrac{1}{2\pi}\displaystyle\int_{-\infty}^\infty e^{ikx}\,dk$
The position eigenstate in position space is: $\langle x|x_0\rangle = \delta(x - x_0)$ — an infinitely sharp spike at $x_0$. This encodes "perfect localization" but, by the Heisenberg uncertainty principle, requires completely indefinite momentum.
Part VII — Momentum Eigenstates
The momentum operator in position space is $\hat{p} = -i\hbar\,d/dx$. Its eigenstates satisfy $\hat{p}|p\rangle = p|p\rangle$, which in position space becomes the differential equation:
This is a plane wave — completely delocalized in position. Normalization gives $\langle p|p'\rangle = \delta(p - p')$. The momentum-space wave function:
This is the Fourier transform of $\psi(x)$! The intimate connection between position and momentum is precisely the Fourier duality — and this is why they obey the Heisenberg uncertainty relation $\Delta x\,\Delta p \geq \hbar/2$.