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The postulates of quantum mechanics assign a pure state to a normalized ray in a complex Hilbert space, and an observable to a self-adjoint operator. A spectral outcome projector $P_a$ has Born rule probability $p_a=⟨\psi |P_a|\psi ⟩$. An ideal projective measurement updates the state to $P_a|\psi ⟩/\sqrt{p_a}$ when $p_a>0$; degeneracy requires the full outcome projector.

Between measurements, an isolated system obeys $i\hslash {\partial }_t|\psi ⟩=H|\psi ⟩$, giving unitary evolution. Composite systems use tensor products; identical particles additionally obey the appropriate exchange symmetry. Mixed preparations use a density matrix, with $p_a=\mathrm{tr}(\rho P_a)$. These postulates require a specified Hamiltonian and measurement model; they do not assign those automatically.^[ref]

For the nonrelativistic Coulomb Hamiltonian with reduced electron-proton mass $\mu$, separation in Spherical coordinate system gives ${\psi }_{n\ell m}(r,\theta ,\varphi )=R_{n\ell }(r)Y_{\ell m}(\theta ,\varphi )$. The quantum numbers obey $n\ge 1$, $0\le \ell \le n-1$ and $-\ell \le m\le \ell$. These are joint eigenstates of energy, $L^2$ and $L_z$, with the latter two eigenvalues $\ell (\ell +1){\hslash }^2$ and $m\hslash$.

The Coulomb bound-state energies and characteristic length areFor example, ${\psi }_{100}=(\pi a_{\mu }^3{)}^{-1/2}{e}^{-r/a_{\mu }}$ is spherically symmetric and has no radial node. In general the radial part contains an exponential times a Generalized Laguerre polynomial, with $n-\ell -1$ radial nodes. The spherical harmonic specifies angular structure and azimuthal phase; real linear combinations of degenerate $m$ states give familiar directional orbitals.

The leading Coulomb energy depends on $n$ alone, giving $n^2$ spatial states for each principal shell, or $2n^2$ including electron spin. Fine structure, Lamb shift and Hyperfine structure lift parts of this ideal degeneracy. An atomic orbital is a probability amplitude, not a classical planetary path.^[ref]

For the Quantum harmonic oscillator defineThe Canonical commutation relation gives $[\hat{a},{\hat{a}}^{†}]=1$, andThus the creation operator and annihilation operator raise or lower the energy by one quantum. Positivity of ${\hat{a}}^{†}\hat{a}$ and termination of downward stepping give $E_n=\hslash \omega (n+1/2)$, $n=0,1,\dots$. Normalized actions are $\hat{a}|n⟩=\sqrt{n}|n-1⟩$ and ${\hat{a}}^{†}|n⟩=\sqrt{n+1}|n+1⟩$. The annihilated ground state obeysThe nonzero Ground state energy is the zero-point energy, consistent with position-momentum quantum uncertainty. Since $\hat{x}=\sqrt{\hslash /(2m\omega )}(\hat{a}+{\hat{a}}^{†})$, a perturbation linear in position connects only neighbouring number states. The algebra is especially useful for matrix elements and selection rules.^[ref]

For hyperfine coupling for a spin-one nucleus, let the electron spin be $s=1/2$ and the nuclear spin $I=1$. First couple $\mathbf{j}=\mathbf{L}+\mathbf{s}$, giving $j=L\pm 1/2$, then $\mathbf{F}=\mathbf{j}+\mathbf{I}$, giving $F=|j-I|,|j-I|+1,\dots ,j+I$.

For each $F$, the projection is $m_F=-F,-F+1,\dots ,F$; electron projections similarly satisfy $m_j=-j,\dots ,j$. Summing $2F+1$ gives 18 states for $L=1$ and 30 for $L=2$, agreeing with $(2L+1)(2s+1)(2I+1)$.

Notation qualification: if the printed leading “2” denotes the principal quantum number, a hydrogenic $2d$ bound state does not exist because $L\le n-1$; a physical $d$ state needs $n\ge 3$. If a doublet term symbol ${}^{2}P$ or ${}^{2}D$ was intended, its leading 2 instead means $2s+1$, and the angular-momentum table applies without fixing $n$. The answers above use the explicitly specified $L$ values, while retaining this ambiguity.^[ref]

The classical minimum is $E_{\mathrm{cl}}=0$, attained at $x=p=0$. The quantum oscillator has $E_0=\hslash \omega /2$, its zero-point energy; simultaneous zero position and momentum uncertainties are forbidden.

Let $\ell =\sqrt{\hslash /(m\omega )}$ and $\xi =x/\ell$. The first three normalized harmonic oscillator eigenfunctions areThey have respectively zero, one and two nodes, with even, odd and even parity. Their phases may be reversed without changing the state.^[ref]

In the position representation, the momentum operators are ${\hat{p}}_x=-i\hslash {\partial }_x$ and ${\hat{p}}_y=-i\hslash {\partial }_y$. Thereforesince ${\partial }_{\varphi }=-y{\partial }_x+x{\partial }_y$ at fixed radius. Solving the eigenvalue equation on the circle and imposing $2\pi$ periodicity givesTheir normalization uses the angular measure $d\varphi$ on $[0,2\pi )$.

For the half-ring sine-squared angular state, write $C=\sqrt{8/(3\pi )}$. The stated wavefunction is normalized because $C^2{\int }_0^{\pi }{\sin }^4\varphi d\varphi =1$. It and its first derivative match continuously to zero at the boundaries, so the integrations by parts have no boundary contributions. The angular momentum expectation and second moment areHence $\mathrm{Var}(L_z)=4{\hslash }^2/3$. The jumps in the second derivative do not invalidate this result: the periodic wavefunction has a square-integrable second derivative and a continuous first derivative.

The probability of $L_z=\hslash$ is the squared projection onto $u_1$:The odd sine term gives the imaginary overlap; a real wavefunction need not have zero probability of nonzero angular momentum. Its positive and negative angular-momentum probabilities are equal.^[ref]