Physic Labs

Quantum mechanics

Dirac formalism: quantum states and two-level systems

Bra–ket notation compactly represents states and measurements; a two-level system makes coherent evolution and Rabi oscillations explicit.

In Dirac notation, the ket ∣ψ⟩|\psi\rangle is a state vector, the bra ⟨ψ∣\langle\psi| its dual, and ⟨ϕ∣ψ⟩\langle\phi|\psi\rangle a probability amplitude. The notation is independent of whether states are represented by wavefunctions or finite-dimensional vectors.

iℏddt∣ψ(t)⟩=H∣ψ(t)⟩,P(a)=∣⟨a∣ψ⟩∣2i\hbar\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle,\qquad P(a)=|\langle a|\psi\rangle|^2

Definition: Two-level system

A two-level system has basis states ∣0⟩,∣1⟩|0\rangle,|1\rangle. Under the resonant Hamiltonian H=ℏΩσx/2H=\hbar\Omega\sigma_x/2, an initial ∣0⟩|0\rangle evolves to cos⁡(Ωt/2)∣0⟩−isin⁡(Ωt/2)∣1⟩\cos(\Omega t/2)|0\rangle-i\sin(\Omega t/2)|1\rangle, so the probability of finding ∣1⟩|1\rangle is sin⁡2(Ωt/2)\sin^2(\Omega t/2).

Vary the coupling and detuning to observe the transition probability over time.

Matrix representation

In the ∣0⟩,∣1⟩|0\rangle,|1\rangle basis, a ket is a two-component column and an operator is a matrix. A common driven Hamiltonian is H=ℏ(Δσz+Ωσx)/2H=\hbar(\Delta\sigma_z+\Omega\sigma_x)/2. Detuning Δ\Delta tilts the rotation axis; the generalized Rabi frequency is ΩR=Ω2+Δ2\Omega_R=\sqrt{\Omega^2+\Delta^2}.

Example: Example: transition probability

On resonance, at t=π/(2Ω)t=\pi/(2\Omega) an initial ∣0⟩|0\rangle has probability sin⁡2(π/4)=1/2\sin^2(\pi/4)=1/2 of being in ∣1⟩|1\rangle. A pulse of duration t=π/Ωt=\pi/\Omega transfers it fully to ∣1⟩|1\rangle.

Solution

Substituting Ωt/2=π/4\Omega t/2=\pi/4 into P1=sin⁡2(Ωt/2)P_1=\sin^2(\Omega t/2) gives P1=1/2P_1=1/2.

In Dirac's formalism, states are vectors in Hilbert space and observables are Hermitian operators. For a normalized ∣ψ⟩|\psi\rangle, the expectation is ⟨A⟩=⟨ψ∣A∣ψ⟩\langle A\rangle=\langle\psi|A|\psi\rangle; expansion in the eigenbasis of AA yields outcome probabilities. Tensor products describe composite systems, while density matrices handle mixed states or subsystems whose surroundings are ignored. Bra-ket notation is compact, but operator order still matters because operators generally do not commute.

Quick check

On resonance under H=ℏΩσx/2, how does the transition probability vary?

What resonant pulse duration transfers the initial state completely to the other level?

References

  1. David J. Griffiths, Darrell F. Schroeter (2018). Introduction to Quantum Mechanics