Time moves states, or operators

Science · Quantum & Modern Physics · ages 22-24

Name ______________________ Date ____________

Which property of reversible linear evolution preserves total probability for every normalized initial state?
- Linearity alone
- Unitarity
- Real matrix entries alone
For a closed system with time-independent, self-adjoint H, which operator evolves its state from zero to t? Keep hbar explicit.
- exp(-i H t/hbar)
- exp(-i H² t/hbar)
- exp(+H t/hbar)
In the Heisenberg picture, the state stays at its initial value and observables carry the evolution-related time dependence.
Circle one: True False
H|0>=0 and H|1>=Δ|1>, with orthonormal states. Initially psi0=(|0>+3|1>)/sqrt(10). At Δt/hbar=pi, what is U(t)psi0?
- (|0>+3|1>)/sqrt(10)
- (|0>-3|1>)/sqrt(10)
- (3|0>-|1>)/sqrt(10)
A normalized superposition evolves under time-independent H. Which statement holds in a fixed orthonormal energy eigenbasis?
- Every measurement basis has fixed outcome probabilities.
- Each energy coefficient has a fixed complex phase.
- Each energy-basis population remains fixed.
For arbitrary unitary U and observable A in a finite-dimensional system, psi_S=U psi0. Which expression always equals <psi_S|A|psi_S> in the Heisenberg picture?
- <psi0|U A U†|psi0>
- <psi0|U† A U†|psi0>
- <psi0|U† A U|psi0>
H is time independent. A finite-dimensional observable A has no explicit time dependence and [H,A]=0. What follows in the Heisenberg picture?
- A_H(t)=A, so its expectation is constant.
- A_H(t)=0, so every measured value must be zero.
- A_H(t)=t A, so every expectation grows with time.
H|0>=0 and H|1>=epsilon|1>, with orthonormal states. Initially psi0=|+>=(|0>+|1>)/sqrt(2). At epsilon t/hbar=pi, a student adds the energy probabilities and reports P(+)=1. What is P(+)?
- 1/2
- 0
- 1

Answer key

For grown-ups. Fold this page away before handing over the rest.

Time moves states, or operators W1-mt_3JsTc9VYj8-s1

Unitarity · Unitarity gives U†U=I, so the evolved squared norm equals the initial squared norm.
exp(-i H t/hbar) · The operator is U(t)=exp(-i H t/hbar). The hbar factor makes its exponent dimensionless.
True · The state is |psi0>, while observables transform as A_H(t)=U†(t) A U(t).
(|0>-3|1>)/sqrt(10) · The zero-energy component gains factor one. The other gains exp(-i pi)=-1, giving (|0>-3|1>)/sqrt(10).
Each energy-basis population remains fixed. · Each coefficient gains a phase factor of magnitude one. Its population stays fixed, although relative phases can change other probabilities.
<psi0|U† A U|psi0> · The evolved bra is <psi0|U† and the evolved ket is U|psi0>. Substitution gives <psi0|U† A U|psi0>.
A_H(t)=A, so its expectation is constant. · A commutes with U(t), so U† A U=U†U A=A. Its expectation in the fixed state is constant and need not be zero.
0 · The state becomes (|0>-|1>)/sqrt(2), whose overlap with |+> is zero. Thus P(+)=0. Adding energy probabilities checks normalization, not this outcome.