Unitary Time Evolution and the Heisenberg Picture
A closed system with a self-adjoint Hamiltonian has unitary time evolution, preserving state norms and total probability. For time-independent H, U(t)=exp(-i H t/hbar). The Heisenberg picture transfers evolution-related time dependence to observables and gives the same measurement predictions.
What a learner can do afterwards
- Writes the evolution operator for a time-independent Hamiltonian and applies it to a superposition
- Explains why unitarity is exactly what conserves total probability
- Moves a problem between the Schrödinger and Heisenberg pictures and checks the answers agree
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For a closed system with time-independent, self-adjoint Hamiltonian H, U(t)=exp(-i H t/hbar). Here i²=-1 and hbar is Planck's constant divided by 2 pi. Take the starting time as zero. The Schrödinger-picture state is |psi_S(t)>=U(t)|psi0>. The exponent is dimensionless. Omitting hbar would require a stated convention hbar=1.
In an orthonormal energy eigenbasis, H|n>=E_n|n>. Each coefficient evolves as c_n(t)=c_n exp(-i E_n t/hbar). For real theta, |exp(-i theta)|=1; exp(-i pi)=-1 and exp(-i pi/2)=-i. Thus energy-basis populations |c_n|² stay fixed. For a degenerate energy, add its basis-state populations to get that energy's probability. Relative phases can change probabilities in another measurement basis.
Unitarity means U†U=UU†=I. The dagger denotes the adjoint and I is the identity. Then <Upsi|Upsi>=<psi|U†U|psi>=<psi|psi>. Every vector keeps its original norm, which need not be one. A state's squared norm is total probability, so a normalized state stays normalized. For reversible linear evolution, preserving every vector's norm is equivalent to unitarity.
For the finite-dimensional examples here, let A have no explicit time dependence. The Heisenberg state is |psi_H>=|psi0> and A_H(t)=U†(t) A U(t). Substituting |psi_S>=U|psi0> and <psi_S|=<psi0|U† gives <psi_S|A|psi_S>=<psi0|A_H|psi0>. Transforming measurement projectors the same way gives equal outcome probabilities in both pictures.
Let H|0>=0 and H|1>=epsilon|1>, with orthonormal states. Start in |+>=(|0>+|1>)/sqrt(2). At epsilon t/hbar=pi/2, psi(t)=(|0>-i|1>)/sqrt(2). To find P(+), square the magnitude of the overlap: P(+)=|<+|psi(t)>|²=|(1-i)/2|²=1/2. Initially it was one. Energy populations remain 1/2 each, but this measurement probability changes.
For fixed H and a finite-dimensional observable A with no explicit time dependence, dA_H/dt=(i/hbar)[H,A_H]. Here [H,A_H]=H A_H-A_H H. If [H,A]=0, A commutes with U, so A_H=U† A U=U†U A=A. Its expectation is constant. This can help with conservation questions, but neither picture always uses less algebra. An explicitly time-dependent A would add another derivative term.
U(t)=exp(-i H t/hbar) preserves norms. Relative phases can change measurement probabilities. A_H=U† A U gives the same predictions with a fixed state.
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