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An energy eigenstate is a quantum state represented by an eigenvector of the Quantum Hamiltonian. For a time-independent Quantum Hamiltonian, a normalized eigenfunction $\varphi$ with energy eigenvalue $E$ evolves according to the Schrödinger equation asHere $\hslash =h/(2\pi )$ is the reduced Planck constant. The changing factor is a global phase, so this is a stationary state: its position probability density is constant. For example, the second infinite square well mode has two density lobes separated by a node of a standing wave at $x=L/2$; both lobes stay fixed while its quantum phase rotates. An ideal energy measurement returns $E$ with probability one, but a position measurement still has a distribution. Stationary density does not describe a particle sitting at the peak of a lobe. Distinct energy eigenvalues accumulate different quantum phases, making their quantum superposition behave differently.

The Born rule uses the squared modulus of the wavefunction, not its real part. Write $\psi =A[\sin x+i(1-\cos x)]$. ThenWavefunction normalization requiresIntegrating the probability density over the required interval gives the dimensionless probabilityThus approximately $9.08\%$ of position measurements fall in this interval.

For a position measurement, the Born rule integrates the probability density over the chosen interval. The three terms from part (b) have integralsandThusThe oscillating term comes from quantum interference between the zero and positive energy components. The probability lies between $1/8-1/(3\pi )$ and $1/8+1/(3\pi )$, both within $[0,1]$; integrating the same probability density over the whole circle gives one.

Assume the given wavefunction is zero outside $[0,L]$. Wavefunction normalization requiresAn arbitrary overall quantum phase is unobservable, so choose the positive real representative $N=\sqrt{3}/L^{3/2}$.

The linear ramp state in an infinite square well has a boundary subtlety: the limiting value at $L$ is $\sqrt{3/L}$, rather than zero. The function is still a normalized element of the well's Hilbert space $L^2(0,L)$, so position measurement and energy-projection probabilities are meaningful. It is not in the domain of the Dirichlet Hamiltonian, nor is it a finite-energy box state. Altering the value at the single endpoint does not cure its nonzero boundary trace; a physical finite-energy preparation must smooth the function to zero near the wall. The following probabilities refer to the supplied idealized state.

An ideal position measurement of a normalized wavefunction $\psi$ assigns a probability to each spatial region $A$ through the Born rule: $P(A)={\int }_A|\psi (x){|}^{2}dx$. This specifies outcome statistics; it does not by itself specify the post-measurement state. A physical detector has finite spatial resolution, which must be included in its measurement model.

The position operator represents a position measurement. In the position representation in quantum mechanics, its one-dimensional action is multiplication, $(X\psi )(x)=x\psi (x)$. The Born rule associates an interval with the integral of $|\psi (x){|}^2$ over that interval, rather than assigning a finite probability to a single point of a continuous distribution.

The expectation value is $⟨X⟩=\int x|\psi (x){|}^{2}dx$ when that integral exists. Its domain in the Hilbert space of square-integrable wavefunctions requires $x\psi (x)$ to be square integrable; a normalized state need not belong to this domain. Together with the momentum operator, the position operator obeys the canonical commutation relation on a suitable common domain, leading to the Heisenberg uncertainty principle.

For a normalized wavefunction $\psi (x)$, the probability of a position measurement lying in a measurable set $A$ is ${\int }_A|\psi (x){|}^{2}dx$. It is a spatial-projection probability and differs from an energy measurement outcome probability.