Operators, Eigenvalues and Measurement
Every observable is represented by an operator, and the values a measurement can return are that operator's eigenvalues. A state that is not an eigenstate gives one of those values with a probability set by the wavefunction.
What a learner can do afterwards
- Applies the position, momentum and energy operators to a stated wavefunction
- Identifies eigenstates and reads off the eigenvalue
- Explains why a superposition returns one of several results rather than an average
1 · Read
Operators turn physical quantities into actions on the wavefunction. Position multiplies by x, momentum differentiates in space scaled by a constant, and energy drives time change through the Hamiltonian. The Schrodinger equation itself is an energy eigenvalue problem: H acting on psi returns E times psi. Each solution pairs one pattern with one energy, like the even oscillator ladder you met.
Test any wavefunction with the apply-and-compare move. Apply the operator and check what comes back. Getting the same function times a number means you hold an eigenstate, and the number is the eigenvalue. If H applied to psi returns 5 eV times psi, then psi is an energy eigenstate with eigenvalue 5 eV. Anything else means it is not an eigenstate.
Measurement acts on the system. A system in an eigenstate of the measured operator returns its eigenvalue with certainty. A superposition returns one eigenvalue at random, with odds given by the squared amplitudes, and the state settles onto the winner. Averages over many trials give the expectation value. No pair like position and momentum shares an eigenstate, so sharpening one blurs the other.
For measurement questions, ask eigenstate of what, then apply and compare. Never average eigenvalues into a single result: one trial gives one eigenvalue. A single result says nothing about the past, since repeats follow the odds and average to the expectation value.
Operators act on wavefunctions, eigenstates return eigenvalues, and superpositions yield one random eigenvalue per trial with odds from squared amplitudes.
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Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.