Eigenvalues and expectation values
PHYS 410 · Quantum Formalism
Eigenvalues are possible measurement results, while expectation values are statistical averages. This lesson explains spectra, probability distributions, variance, and repeated measurements.
Key equations
hat{A}|a_n
angle=a_n|a_n
angle|psi
angle=sum_n c_n|a_n
angleP(a_n)=|c_n|^2sum_n |c_n|^2=1psi(x)=langle x|psi
angleP(x)=|psi(x)|^2P(aleq xleq b)=int_a^b |psi(x)|^2dxlangle A
angle=sum_n a_n |c_n|^2langle A
angle=langlepsi|hat{A}|psi
anglelangle x
angle=int x|psi(x)|^2dxlangle p
angle=int psi^*(x)left(-ihbar rac{d}{dx}
ight)psi(x)dxsigma_A^2=langle A^2
angle-langle A
angle^2Delta A=sqrt{langle A^2
angle-langle A
angle^2}Delta A=0 rac{dlangle x
angle}{dt}= rac{langle p
angle}{m}Learning objectives
- Interpret eigenvalues as possible measurement outcomes.
- Calculate probabilities from expansion coefficients.
- Compute expectation values in discrete and continuous cases.
- Calculate variance and uncertainty.
- Distinguish expectation values from individual measurement outcomes.
Measurement results and eigenvalues
In quantum mechanics, the possible results of measuring an observable are the eigenvalues of the operator representing that observable. If
angle=a_n|a_n angle$$ then $a_n$ is a possible measurement result. If the system is already in eigenstate $|a_n angle$, then measurement of $A$ gives $a_n$ with probability 1. If the state is a superposition, outcomes are probabilistic. ## Discrete spectra For observables with discrete eigenvalues, a normalized state can be written as $$|psi angle=sum_n c_n|a_n angle$$ The probability of outcome $a_n$ is $$P(a_n)=|c_n|^2$$ with $$sum_n |c_n|^2=1$$ Bound-state energies, spin components, and harmonic oscillator levels often have discrete spectra. ## Continuous spectra Some observables, such as position for a free particle, have continuous spectra. A state in the position basis is described by $$psi(x)=langle x|psi angle$$ and the probability density is $$P(x)=|psi(x)|^2$$ The probability for an interval is $$P(aleq xleq b)=int_a^b |psi(x)|^2dx$$ ## Expectation value The expectation value is the average result over many identically prepared measurements. For a discrete observable, $$langle A angle=sum_n a_n |c_n|^2$$ In operator form, $$langle A angle=langlepsi|hat{A}|psi angle$$ For position, $$langle x angle=int x|psi(x)|^2dx$$ For momentum in position space, $$langle p angle=int psi^*(x)left(-ihbar rac{d}{dx} ight)psi(x)dx$$ ## Variance and uncertainty The spread of measurement outcomes is described by variance: $$sigma_A^2=langle A^2 angle-langle A angle^2$$ The uncertainty is the standard deviation: $$Delta A=sqrt{langle A^2 angle-langle A angle^2}$$ If a state is an eigenstate of $hat{A}$, then $Delta A=0$ for that observable. ## Expectation is not always observed An expectation value need not be a possible measurement outcome. For a spin-1/2 particle, measuring $S_z$ gives either $+hbar/2$ or $-hbar/2$, but the expectation value can be zero. This means expectation values describe ensemble averages, not necessarily individual events. ## Time dependence Expectation values can change in time as the state evolves. Ehrenfest's theorem shows that expectation values often obey equations resembling classical mechanics. For example, $$ rac{dlangle x angle}{dt}= rac{langle p angle}{m}$$ for common Hamiltonians. This helps connect quantum mechanics to classical motion. ## The big idea Eigenvalues are possible outcomes of single measurements. Expectation values are statistical averages over many measurements. Variance and uncertainty quantify spread. Quantum theory predicts probability distributions, not generally definite preexisting values for every observable.Ask your AI physics guide
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