Uncertainty Relations

The uncertainty principle is the quantitative statement that non-commuting observables cannot both have sharp values in the same state. It is not a statement about measurement disturbance or experimental imperfection; it is a theorem about the spread of the probability distributions assigned by a state $|\psi ⟩$ through the Born rule. This page derives the general Robertson–Schrödinger inequality, specializes it to position and momentum, discusses the (frequently misstated) energy–time relation, and identifies the minimum-uncertainty states.

1. Variance of an Observable

For an observable $\hat{A}$ (Hermitian) and a normalized state $|\psi ⟩$, define the uncertainty $\Delta A$ as the standard deviation of its measurement outcomes:

$$(\Delta A{)}^2=⟨{\hat{A}}^2⟩-⟨\hat{A}{⟩}^2=⟨(\hat{A}-⟨\hat{A}⟩{)}^2⟩.$$

Writing $\delta \hat{A}=\hat{A}-⟨\hat{A}⟩\hat{1}$, we have $(\Delta A{)}^2=⟨\delta {\hat{A}}^2⟩=\Vert \delta \hat{A}|\psi ⟩{\Vert }^2$. The uncertainty vanishes iff $|\psi ⟩$ is an eigenstate of $\hat{A}$.

2. The Robertson–Schrödinger Inequality

Let $|f⟩=\delta \hat{A}|\psi ⟩$ and $|g⟩=\delta \hat{B}|\psi ⟩$. The Cauchy–Schwarz inequality $\Vert f{\Vert }^2\Vert g{\Vert }^2\ge |⟨f|g⟩{|}^2$ gives

$$(\Delta A{)}^2(\Delta B{)}^2\ge |⟨\delta \hat{A}\delta \hat{B}⟩{|}^2.$$

Split the operator product into its Hermitian and anti-Hermitian parts,

$$\delta \hat{A}\delta \hat{B}=\frac{1}{2}\{\delta \hat{A},\delta \hat{B}\}+\frac{1}{2}[\hat{A},\hat{B}],$$

where $\{\cdot ,\cdot \}$ is the anticommutator. The first term has real expectation, the second (times $1/i$) real expectation, so the two contributions add in quadrature:

$$\boxed{(\Delta A{)}^2(\Delta B{)}^2\ge (\frac{1}{2}⟨\{\delta \hat{A},\delta \hat{B}\}⟩{)}^2+(\frac{1}{2i}⟨[\hat{A},\hat{B}]⟩{)}^2.}$$

This is the Schrödinger form. Dropping the (nonnegative) anticommutator term gives the weaker, more familiar Robertson relation:

$$\Delta A\Delta B\ge \frac{1}{2}|⟨[\hat{A},\hat{B}]⟩|.$$

3. Position–Momentum

Using the canonical commutator $[\hat{x},\hat{p}]=i\hslash$, the Robertson relation gives the Heisenberg uncertainty principle:

$$\boxed{\Delta x\Delta p\ge \frac{\hslash }{2}.}$$

Because $\psi (x)$ and $\tilde{\psi }(p)$ are a Fourier-transform pair, this is exactly the classical bandwidth theorem: a function and its transform cannot both be arbitrarily narrow. Localizing a particle in space forces a broad momentum distribution, and vice versa.

4. Minimum-Uncertainty (Gaussian) States

Equality in Cauchy–Schwarz requires $|g⟩\propto |f⟩$, i.e. $\delta \hat{p}|\psi ⟩=\lambda \delta \hat{x}|\psi ⟩$, and equality in §2 additionally requires the anticommutator term to vanish, forcing $\lambda$ pure imaginary. Solving the resulting first-order ODE in the position representation gives a Gaussian wavepacket,

$$\psi (x)\propto \exp \left[-\frac{(x-⟨x⟩{)}^2}{4(\Delta x{)}^2}+\frac{i⟨p⟩x}{\hslash }\right],$$

the unique states saturating $\Delta x\Delta p=\hslash /2$. These are precisely the coherent states of the harmonic oscillator, the "most classical" quantum states.

5. The Energy–Time Relation

The relation

$$\Delta E\Delta t\gtrsim \frac{\hslash }{2}$$

is not an instance of §2, because time is a parameter in non-relativistic QM, not an operator (there is no Hermitian $\hat{t}$ conjugate to $\hat{H}$). Its correct reading (Mandelstam–Tamm): for any observable $\hat{B}$, define the characteristic evolution time $\Delta t_B=\Delta B/|d⟨\hat{B}⟩/dt|$ — the time for $⟨\hat{B}⟩$ to change by one standard deviation. Then the Robertson relation applied to $\hat{H}$ and $\hat{B}$ yields

$$\Delta E\Delta t_B\ge \frac{\hslash }{2}.$$

So $\Delta t$ is the timescale on which the state changes appreciably, not an uncertainty in a measured time. Consequences: stationary states ($\Delta E=0$) never evolve; short-lived states (small $\Delta t$) have broad energy widths — the origin of the natural linewidth of the decay processes treated by time-dependent perturbation theory.

See also

• Wave Mechanics — Fourier duality behind $\Delta x\Delta p$.
• Harmonic Oscillator — coherent states as minimum-uncertainty states.
• Postulates — the Born rule that defines the distributions being bounded.