General Formalism

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

A central potential is a potential that depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant. We may use these properties to reduce this otherwise three-dimensional problem to an effective one-dimensional problem. The general form of the Hamiltonian for a particle immersed in such a potential is

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(|{\hat {r}}|).$

Due to rotational symmetry, $[{\hat {H}},{\hat {L}}_{z}]=0\!$ and $[{\hat {H}},{\hat {L}}^{2}]=0.\!$ This allows us to find a complete set of states that are simultaneous eigenstates of ${\hat {H}},\!$ ${\hat {L}}_{z},\!$ and ${\hat {L}}^{2}.\!$ We will label these eigenstates as $|n,l,m\rangle ,\!$ where $l\!$ and $m\!$ are as defined in the previous chapter and $n\!$ represents the quantum numbers that define the radial dependence of the wave function; this is the only part of the state that depends on the exact form of the potential, as we will see shortly.

From this we can get a state of the same energy for a given $l\!$ with a degeneracy of $2l+1\!$. We can rewrite the Laplacian as

$\nabla ^{2}={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {L^{2}}{\hbar ^{2}r^{2}}}$

This makes the Schrödinger equation

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {L^{2}}{2mr^{2}}}+V(r)\right)\psi (r,\theta ,\phi )=E\psi (r,\theta ,\phi )$

Using separation of variables, $\psi (r,\theta ,\phi )=f_{l}(r)Y_{lm}(\theta ,\phi )\!$, we get:

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)f_{l}(r)Y_{lm}(\theta ,\phi )=Ef_{l}(r)Y_{lm}(\theta ,\phi )$

The term ${\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}$ is referred to as the centrifugal barrier, which is associated with the motion of the particle. The classical analogue is ${\frac {l^{2}}{2mr^{2}}}$. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point. Multiplying both sides by $Y_{l^{\prime }m'}\!$ and integrating over the angular dependence reduces the equation to merely a function of $r\!$.

Now if we let $u_{l}(r)=rf_{l}(r)\!$, this gives the radial Schrödinger equation:

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)u_{l}(r)=Eu_{l}(r)$

Due to the boundary condition that $f_{l}(r)\!$ must be finite the origin, $u_{l}(r)\!$ must vanish.

Often looking at the asymptotic behavior of $u_{l}(r)\!$ can be quite helpful.

As $r\rightarrow 0\!$ and $V(r)\ll {\frac {1}{r^{2}}}\!$, the dominating term becomes the centrifugal barrier giving the approximate Hamiltonian:

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}$

which has the solutions $u_{l}(r)\sim r^{l+1},r^{-l}\!$ where only the first term is physically possible because the second blows up at the origin.

As $r\rightarrow \infty \!$ and $rV(r)\rightarrow 0$(which does not include the monopole ${\frac {1}{r}}$ coulomb potential), the Hamiltonian approximately becomes

$-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial r^{2}}}u_{l}(r)=Eu_{l}(r)$.

Letting $k=-i{\sqrt {\frac {2mE}{\hbar ^{2}}}}$ gives a solution of $u_{l}(r)=Ae^{kr}+Be^{-kr}\!$, where when $k\!$ is real, $B=0\!$, but both terms are needed when $k\!$ is imaginary.

Nomenclature

Historically, the first four (previously five) values of $l\!$ have taken on names, and additional values of $l\!$ are referred to alphabetically:

${\begin{cases}l=0&{\mbox{s-wave (sharp)}}\\l=1&{\mbox{p-wave (principal)}}\\l=2&{\mbox{d-wave (diffuse)}}\\l=3&{\mbox{f-wave (fundamental)}}\\l=4&{\mbox{g-wave (previously called t-wave for thick)}}\\l=5&{\mbox{h-wave}}\\\end{cases}}$

Worked Problem involving the energy levels in a central potential.