= Solution
The <postulates of quantum mechanics> give a framework for predicting observations from a prepared state. The experiments below motivate different features of that framework; no one experiment proves the whole set of postulates or specifies every <Hamiltonian operator>.
A pure <quantum state> is a normalized vector in a complex <Hilbert space>, with vectors differing by an overall <quantum phase> representing the same physical preparation. In <position> representation, $\int|\psi(\mathbf r)|^2d^3r=1$. The linear state space permits <quantum superposition>, while the <Born rule> assigns <probability> $\int_\Omega|\psi|^2d^3r$ to a <position> outcome in a region $\Omega$. The <wavefunction> is a <probability amplitude>; it is not a classical distribution of definite particle positions and momenta.
An observable is represented by a <self-adjoint operator>. If $Q$ has spectral projectors $P_q$, its possible measurement outcomes are its spectral values and $\Pr(q)=\langle\psi|P_q|\psi\rangle$. The mean is $\langle Q\rangle=\langle\psi|Q|\psi\rangle$. For an ideal repeatable <projective measurement>, obtaining $q$ leaves the normalized conditional state $P_q\psi/\sqrt{\Pr(q)}$. This projects onto an entire eigenspace if the outcome is degenerate. General detectors need more general <measurement in quantum mechanics> models; the projective update is an idealization. Noncommuting observables need not have simultaneous sharp values, as the <canonical commutation relation> and <Heisenberg uncertainty relation> illustrate.
Between measurements, an isolated system has <unitary time evolution> generated by its <Hamiltonian operator>:
$$
i\hbar\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle,\qquad U(t)=e^{-iHt/\hbar}\quad\text{for fixed }H.
$$
This preserves normalization and superpositions. Composite systems use a <tensor product> of state spaces, allowing entangled states as well as product states; identical particles additionally require the bosonic or fermionic exchange symmetry appropriate to their species. Statistical mixtures are described by a <density matrix>, with $\Pr(q)=\operatorname{tr}(\rho P_q)$. These rules separate preparation, coherent evolution and measurement.
The <photoelectric effect> exposes discrete <energy> transfer by light. Illuminate a clean metal cathode in an evacuated tube and measure the collected <Electron> current while varying the light <frequency>, intensity and a retarding <voltage>. The <stopping potential> $V_s$ probes the largest emitted <Electron> <kinetic energy>, whereas the <photocurrent> measures the rate at which charge is collected. In the ordinary single-photon regime there is a threshold <frequency>; above it the maximum <energy> grows with <frequency>, rather than with intensity. Changing intensity at fixed <frequency> mainly changes the number of emitted electrons. These observations are summarized by the <Einstein photoelectric equation>
$$
\boxed{K_{\max}=h\nu-\Phi=eV_s,\qquad \nu_0=\Phi/h.}
$$
Here $\Phi$ is the surface <work function>, and the stopping <voltage> is understood after any <photoelectric contact-potential correction>. A <photon> delivers $h\nu$ in an absorption event; emission is allowed only when that <energy> can pay the escape cost. This accounts for the <frequency> threshold and the lack of the classical waiting time associated with slowly accumulating <energy> in a single <Electron>. The experiment supports <energy> quantization of light; it does not imply that every incident <photon> emits an <Electron> or that every emitted <Electron> has the maximum <energy>. Surface losses and collection geometry affect the full current-voltage curve.
The <Davisson–Germer experiment> demonstrates a complementary wave property of matter. An <Electron> gun uses a heated emitter, accelerating <voltage> and apertures to send a narrow beam onto a nickel single crystal in vacuum. A movable collector measures scattered <electric current> versus direction; rotating the crystal also tests its azimuthal symmetry. The crystal gives sharp angular beams whose positions depend on <voltage> and crystal orientation. A prominent feature occurs near $54\,\mathrm V$ accelerating <voltage> and roughly $50^\circ$ detector colatitude. The nonrelativistic <de Broglie wavelength> predicted at that <voltage> is
$$
\lambda=\frac h{\sqrt{2m_e eU}}\simeq0.167\,\mathrm{nm}.
$$
<Diffraction> wavelengths inferred using the known crystal structure agree with this <voltage> dependence. These angular peaks arise from coherent addition of scattering amplitudes from the crystal, not from classical specular reflection of independent point projectiles. <Bragg's law> is $2d\sin\theta_B=r\lambda$ for lattice-plane spacing $d$ and angle to those planes; a detector colatitude is a different angle and must not be substituted without its geometry. The historical readout was current, not a photograph of individual <Electron> trajectories.<Diffraction of Electrons by a Crystal of Nickel by C. Davisson and L. H. Germer>[1927, apparatus pp. 707–710 and angular/<voltage> scans pp. 715–719]{ref}
The <double-slit experiment> makes the role of amplitudes especially direct. A coherent source illuminates two narrow slits; an observation screen a distance $D$ away records the pattern. For slit spacing $d$ and small screen displacement $y$, the path difference is approximately $dy/D$. The two amplitudes add before probabilities are formed:
$$
\boxed{P(y)=|\psi_1(y)+\psi_2(y)|^2=|\psi_1|^2+|\psi_2|^2+2\operatorname{Re}(\psi_1^*\psi_2).}
$$
For equal coherent illumination the <interference> term varies as $\cos(2\pi dy/(\lambda D))$, giving fringe spacing $\lambda D/d$ beneath the single-slit <diffraction> envelope. Blocking one slit removes that cross term. Young's optical experiment established wave <interference>; later particle-resolved realizations with photons or electrons show localized detection events whose accumulated distribution has the <interference> pattern. Low arrival rates demonstrate that this pattern does not require different incident particles to collide with one another.
If the paths become correlated with distinguishable detector states $|D_1\rangle$ and $|D_2\rangle$, the cross term acquires their overlap $\langle D_2|D_1\rangle$. Perfectly distinguishable path records make that overlap zero, suppressing <interference> even if the spatial beams still overlap. This connects <quantum entanglement>, measurement and loss of observable coherence. Without a path measurement, the two alternatives are amplitudes for one preparation, rather than classical mutually exclusive events whose probabilities may simply be added.
Together these experiments motivate a description in which <energy> transfer and detector outcomes can be discrete, yet propagation retains complex <quantum phases> and <interference>. The <Born rule> connects those amplitudes to repeatable frequencies of outcomes. The <Schrödinger equation> supplies the dynamics once a <Hamiltonian operator> is specified, while observable operators, state preparation and the measurement model specify what the apparatus can read.
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