Quantum Uncertainty: Derivation, Saturation, and Energy-Time Insights from MIT OCW

MIT OpenCourseWare guides you through a precise formulation of the uncertainty principle for two Hermitian operators A and B. Using the auxiliary states F and G, the video derives the inequality Delta A squared Delta B squared ≥ (1/4) |⟨[A,B]⟩|^2, explains when equality is achieved, and discusses the energy time version by linking the time derivative of an operator to its commutator with the Hamiltonian. An illustrative spin example with SX in a Sz eigenstate and the classic X–P pair are worked out in detail. The talk also highlights a practical application to the hydrogen hyperfine transition, showing how a long lifetime leads to an extremely narrow spectral line, and ends with reflections on the interpretation and limitations of energy time uncertainty.

Introduction to the Uncertainty Principle

The video begins by formalizing uncertainty as the norm of an operator minus its expectation value, Delta A equals ||(A - ⟨A⟩)ψ||, and shows how it vanishes if and only if the state is an eigenstate of A. Delta A^2 is written as ⟨A^2⟩ - ⟨A⟩^2, establishing a concrete route to compute uncertainties for concrete observables.

A Concrete Spin Example

To illustrate the calculation, the speaker considers a state that is an eigenstate of Sz (the Z-basis "plus" state). SX does not commute with Sz, so Delta SX is nonzero even though ⟨SX⟩ = 0 in that state. Using the SX matrix representation, the calculation shows SX^2 is proportional to the identity, yielding Delta SX = ħ/2 for a spin-1/2 system in Sz eigenstate.

Deriving the Uncertainty Principle

The core derivation introduces auxiliary operators F = A − ⟨A⟩ and G = B − ⟨B⟩. By applying Schwarz’s inequality to these two vectors in Hilbert space, the video arrives at Delta A^2 Delta B^2 ≥ |⟨F,G⟩|^2, then rewrites the right-hand side in terms of the commutator and anti-commutator of A and B. This yields the familiar form of the uncertainty principle: Delta A Delta B ≥ (1/2) |⟨[A,B]⟩|, with a real, positive right-hand side that is associated with the commutator A B − B A.

Saturation and Minimum Uncertainty States

The discussion then addresses when the bound can be saturated. Saturation requires F and G to be parallel (up to a complex scalar). This leads to a condition involving the overlap ⟨F|G⟩ and imposes that the complex proportionality constant be purely imaginary. The analysis connects these mathematical conditions to the physical idea of minimum uncertainty states, which can be found by solving a differential equation that emerges from the saturation condition.

Energy-Time Uncertainty

Turning to energy and time, the video derives an energy-time relation by choosing A = H (the Hamiltonian) and B = Q, a time-independent operator. Since there is no explicit time dependence in Q, the time derivative of its expectation value relates to the commutator with H via d⟨Q⟩/dt = (i/ħ)⟨[H,Q]⟩. Substituting back into the uncertainty expression and rearranging yields a form that ties the energy uncertainty Delta E to the rate of change of Q, introducing a time scale Delta T ~ Delta Q / |d⟨Q⟩/dt|, and giving Delta E Delta T ≥ ħ/2.

Applications and Physical Insight

The talk remarks on the non-universality of Delta T, noting that the time uncertainty depends on the chosen operator Q. It also discusses energy conservation in isolated systems with time-independent Hamiltonians, showing that Delta E remains constant even though Q may evolve in time. A physical example is provided through atomic decay, where a long lifetime tau implies a very small energy uncertainty, and the emitted photon carries the energy uncertainty. A celebrated application is the hyperfine transition in hydrogen, which produces the 21 cm line at about 1420 MHz. The extraordinary sharpness of this line is attributed to the lifetime of the upper state being on the order of 10 million years, yielding an extremely small Delta E and a correspondingly narrow spectral width.

Concluding Remarks

The presenter closes by reinforcing the usefulness of the uncertainty principle for bounding energies, understanding saturation states, and connecting these ideas to concrete physical systems such as atomic transitions and spectral lines. The discussion hints at further explorations in recitation on minimum uncertainty wave packets and the broader implications for quantum dynamics.