Diagnose quantum-mechanics readiness

This untimed diagnostic asks whether you can use states, observables, spectra, commutators, symmetry, and Schrödinger and Heisenberg pictures in a small quantum system. It tests visible work, not whether you remember a course title. Use any accessible format that shows the Hilbert space, operators, assumptions, and reasoning; a symbolic tool may check matrix arithmetic after those are set.

All three tasks contribute to one capability result. They also mark an important boundary: success in a finite-dimensional system prepares the operator logic used in QFT, but does not by itself establish field-theoretic claims about distributions, infinite volume, particles, or spectral measures.

States, spectra, and measurement

In a two-level Hilbert space, let

$$\rho=\frac12(1+r\sigma_x),\qquad -1\le r\le1, \qquad A=\mathbf n\mathbin{\cdot}\boldsymbol\sigma, \qquad |\mathbf n|=1.$$

1. Verify that $\rho$ is normalized and positive. Determine when it is pure.
2. Find the eigenvalues and spectral projectors of $A$.
3. Compute the probability of each outcome, verify normalization, and recover $\langle A\rangle_\rho$ from the spectral sum.
4. Compute $\operatorname{Var}_\rho(A)$.
5. In words, distinguish the state, observable, eigenvalue, projector, probability, and expectation value.

Answer guide

The eigenvalues of $\rho$ are $(1\pm r)/2$, so it is positive precisely for $|r|\le1$, and $\operatorname{Tr}\rho=1$. Since $\operatorname{Tr}\rho^2=(1+r^2)/2$, it is pure when $|r|=1$ and mixed when $|r|<1$.

Because $A^2=1$, its eigenvalues are $a=\pm1$ and

$$P_\pm=\frac12(1\pm\mathbf n\mathbin{\cdot}\boldsymbol\sigma).$$

Writing $n_x$ for the $x$ component of $\mathbf n$,

$$p_\pm=\operatorname{Tr}(\rho P_\pm)=\frac12(1\pm r n_x), \qquad p_++p_-=1.$$

Then $\langle A\rangle=p_+-p_-=rn_x$ and, because $A^2=1$, $\operatorname{Var}(A)=1-r^2n_x^2$. The density operator represents the state; $A$ is an observable; its eigenvalues are possible outcomes; the projectors select their eigenspaces; the Born rule assigns state-dependent probabilities; and the expectation is their probability-weighted mean.

Commutators, symmetry, and pictures

Now take

$$H=\frac{\omega}{2}\sigma_z, \qquad |\psi(0)\rangle=\frac{|0\rangle+|1\rangle}{\sqrt2}, \qquad B=\sigma_x, \qquad Q=\frac12\sigma_z,$$

with $\omega>0$ and $\sigma_z|0\rangle=|0\rangle$.

1. Find the spectrum and projectors of $H$.
2. Evolve the state in the Schrödinger picture and calculate $\langle B\rangle(t)$.
3. Evolve $B$ in the Heisenberg picture, keep the state fixed, and calculate the same expectation value.
4. Verify the Heisenberg equation from $[H,B]$.
5. Test whether $Q$ generates a continuous symmetry of $H$ and state the associated conserved quantity.
6. Explain why evolving both the state and the Heisenberg operator in the same matrix element double counts time evolution.

Answer guide

$H$ has eigenvalues $\pm\omega/2$ and projectors $P_0=(1+\sigma_z)/2$, $P_1=(1-\sigma_z)/2$. The Schrödinger state is

$$|\psi(t)\rangle=\frac{e^{-i\omega t/2}|0\rangle +e^{+i\omega t/2}|1\rangle}{\sqrt2},$$

so $\langle B\rangle_S(t)=\cos(\omega t)$. In the Heisenberg picture,

$$B_H(t)=e^{iHt}\sigma_xe^{-iHt} =\sigma_x\cos(\omega t)-\sigma_y\sin(\omega t),$$

and the fixed initial state again gives $\cos(\omega t)$. Since $[\sigma_z,\sigma_x]=2i\sigma_y$, $\dot B_H=i[H,B_H]$ reproduces this rotation. Because $[H,Q]=0$, the unitary $e^{-i\alpha Q}$ is a symmetry and $Q$ is conserved. Schrödinger and Heisenberg pictures redistribute the same unitary evolution; applying both in one matrix element applies that evolution twice.

What transfers to QFT—and what does not

For a finite Hamiltonian with orthonormal energy states $|n\rangle$, insert a complete set into a stationary-state correlator. If $|a\rangle$ has energy $E_a$, then

$$\langle a|B(t)B(0)|a\rangle =\sum_n e^{-i(E_n-E_a)t} \langle a|B|n\rangle\langle n|B|a\rangle.$$

When $B$ is Hermitian, each coefficient is $|\langle n|B|a\rangle|^2\ge0$. Explain which parts of this argument use stationarity, completeness, and Hermiticity. Then name additional assumptions needed before calling the analogous QFT object a positive spectral measure.

Answer guide

Stationarity supplies a definite reference energy and makes the time dependence an energy difference. Completeness gives the sum over intermediate states. Hermiticity turns the paired matrix elements into an absolute square.

In QFT one must additionally specify the vacuum or state, Poincaré covariance when used, the spectrum condition, positivity of the Hilbert-space inner product, completeness including multiparticle continua, and the distributional meaning of local fields. Gauge-fixed fields may live in an auxiliary space where positivity is not manifest. Finite-dimensional success therefore illustrates the algebraic pattern but does not prove a QFT spectral theorem. Later, Spectral Decomposition of Two-Point Functions develops the field-theory statement with its hypotheses.

Interpret the work

• Demonstrated: the state is valid; projectors reconstruct the observable; probabilities normalize; both pictures agree; symmetry is tested through its action or commutator; and the spectral transfer states its additional QFT assumptions.
• Uncertain: the matrix results are plausible, but positivity, projector completeness, picture choice, the symmetry action, or a transfer assumption remains implicit.
• Not yet demonstrated: probabilities fail to normalize; a ket is treated as an observable; spectra are detached from projectors; the two pictures disagree through double evolution; or the finite-system argument is asserted as a QFT theorem.

These labels describe this work product, not you. For either of the last two results, use Quantum states and operators, then repeat the tasks with a Hamiltonian whose axis is not diagonal in the original basis. Return to Core QFT or to the specialist path that requested the check.

References

• John Preskill, Lecture Notes for Physics 229: Quantum Information and Computation, updated 2018, official open PDF.
• Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Volume I: Functional Analysis, revised and enlarged edition, Academic Press, 1980.
• Steven Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995, doi:10.1017/CBO9781139644167.