Orbital angular momentum

Orbital angular momentum is $L=-i\hslash ,x\times \nabla$; in particular $L_3=-i\hslash (x_1{\partial }_{x_2}-x_2{\partial }_{x_1})$.

For a differentiable radial factor and integer $n\ge 0$, the coordinate polynomial obeys $L_3\psi =n\hslash \psi$ and $L_{+}\psi =0$. The identity $L^2=L_{-}{L}_{+}+L_3^2+\hslash L_3$ gives the displayed angular eigenvalue. Lowering gives $L_{-}\psi =-2n\hslash z(x+iy{)}^{n-1}f(r)$, a nonzero eigenfunction with the same angular eigenvalue when $n>0$ and the original state is nonzero. At $n=0$ lowering gives the zero function, which is not an eigenfunction. Negative powers satisfy local differential identities off the axis but are not regular angular states.

For a homogeneous harmonic polynomial $H_l$ of degree $l$ in three dimensions, the Euler theorem for homogeneous functions gives $D{H}_l=l{H}_l$, where $D=\mathbf{x}\cdot \nabla$. The orbital angular momentum identity ${\hat{L}}^2=-{\hslash }^2(r^2\Delta -D^2-D)$ therefore gives ${\hat{L}}^2{H}_l=l(l+1){\hslash }^2{H}_l$. This constructs angular momentum eigenstates directly from polynomials.

Orbital angular momentum operators annihilate a radial multiplier $H(r)$. Their squared sum sends every degree-one homogeneous polynomial to $2{\hslash }^2$ times that polynomial, so a nonzero normalizable state $H(r)(\mathbf{a}\cdot \mathbf{x})$ has angular momentum quantum number $l=1$. With real $H$ and real coefficients, its third-component expected value is zero by angular symmetry, though it need not be an eigenstate of that component.

The magnetic quantum number $m$ labels an eigenvalue of $L_z$: $L_z|\ell ,m⟩=m\hslash |\ell ,m⟩$, with $m=-\ell ,-\ell +1,\dots ,\ell$.

One of the angular momentum eigenstates in a multiplet, distinguished by the magnetic quantum number along a chosen axis. An applied magnetic field can split their energy levels through Zeeman splitting.

For $L_i={\epsilon }_{ik\ell }{X}_k{P}_{\ell }$ and $[X_i,P_j]=i\hslash {\delta }_{ij}$,

The canonical position-momentum commutators imply

The operators $L_{\pm }=L_x\pm i{L}_y$ satisfyThey therefore change the magnetic quantum number $m$ by $\pm 1$ while preserving the total angular-momentum quantum number $\ell$.

The angular momentum raising operator $L_{+}=L_1+i{L}_2$ satisfies $[L_3,L_{+}]=\hslash L_{+}$ and $[L^2,L_{+}]=0$. It preserves the total angular momentum while increasing the third-component magnetic quantum number by one, whenever its result is nonzero. Its adjoint operator is the angular momentum lowering operator $L_{-}$. Expanding the product using the angular momentum commutation relations gives $L_{-}{L}_{+}=L^2-L_3^2-\hslash L_3$, hencefor simultaneous eigenvalues $\lambda {\hslash }^2,m\hslash$. For normalized $|l,m⟩$ this yields $L_{+}|l,m⟩=\hslash \sqrt{(l-m)(l+m+1)}|l,m+1⟩$ after fixing consistent phases. At $m=l$ the result is zero, the upper angular momentum ladder endpoint. For a different quantization axis, cyclically permute the components; for example $L_2+i{L}_3$ raises the $L_1$ projection.

The norm of a raised or lowered angular momentum state must be nonnegative. At the upper or lower endpoint, the relevant ladder operator annihilates the state; the zero vector is not an eigenvector.

Operator ordering gives

In position representation,The radial derivatives cancel after substituting $P=-i\hslash \nabla$ into the Cartesian identity for $L^2$.

The coordinate combinations $x_1\pm i{x}_2$ have $L_3$ eigenvalues $\pm \hslash$, so their powers carry magnetic quantum numbers $\pm m$.

If $\psi =z{e}^{-r^2/(2a^2)}$ in two dimensions, then $L_z\psi =\hslash \psi$ and $L_z{\psi }^{\ast }=-\hslash {\psi }^{\ast }$. Orthogonality therefore makes the angular-momentum expectation of $\alpha \psi +\beta {\psi }^{\ast }$ equal to $(|\alpha {|}^2-|\beta {|}^2)\hslash$ for normalized coefficients.

Forboth $P^2$ and $|X{|}^2$ are rotational scalars, so $[H,L_i]=0$. Consequently $H,L^2,L_3$ commute pairwise and can be simultaneously diagonalized.

An energy and orbital angular momentum eigenstate for a real two-dimensional central potential can be chosen with integer $k$ and real radial function $f$. Its radial equation isQuantum degeneracy permits other stationary states, including $f(r)\cos (k\varphi )/\sqrt{\pi }$, which are not single angular exponentials. Separation of variables supplies a basis rather than describing every superposition in a degenerate eigenspace.