Hydrogen Atom

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\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

The Schrödinger equation for the particle moving in central potential can be represented in a spherical coordinate system as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( -\frac{\hbar^2}{2\mu}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{L^2}{2\mu r^2}+V(r)\right) \psi(r,\theta,\phi)=E\psi(r,\theta,\phi)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\!} is the angular momentum operator and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu\!} is the reduced mass.

In this case, being invariant under the rotation, the Hamiltonian, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} , commutes with both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} . Furthermore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} commute with each other. Therefore, the energy eigenstates can be chosen to be simultaneously the eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H\!} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} . Such states can be expressed as the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(r,\theta,\phi)=R(r)Y_{lm}(\theta,\phi)=\frac{u_l(r)}{r}Y_{lm}(\theta,\phi)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y_l^m(\theta,\phi)} is the spherical harmonic, which is the simultaneous eigenstates of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_z\!} , and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{u_l(r)}{r}} substitution is made for simplification.

Substituting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(r,\theta,\phi)=\frac{u_l(r)}{r}Y_{lm}(\theta,\phi) } into the Schrödinger equation, and taking into account the fact that:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2\psi(r,\theta,\phi)=L^2[R(r)Y_{lm}(\theta,\phi)]=R(r)L^2Y_{lm}(\theta,\phi)=R(r)\hbar^2l(l+1)Y_{lm}(\theta,\phi) =\hbar^2l(l+1)\psi(r,\theta,\phi)}

we have the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\!} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V(r)\right] u_l(r)=Eu_l(r)}

In the hydrogen atom or single electron ion, the potential is the Coulomb potential between the electron and the nucleus:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)=-\frac{Ze^2}{r}}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=1\!} for the hydrogen, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=2\!} for helium ion Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle He^+\!} , etc.

Plugging this potential into the Schrödinger equation we obtain

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{Ze^2}{r}\right] u_l(r)=Eu_l(r)}

Since we are only concentrating on the bound states, we can write down the solutions to this equation in the asymptotic limits.

In the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \to 0 } limit the equation becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}\right] u_l(r)=0}

and the solution in this limit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\rightarrow r^{l+1}} .

In the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r \to \infty } limit the equation becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2 u_l(r)}{\partial r^2}+\frac{2mE}{\hbar^2}u_l(r)=0}

and the solution in this limit is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\rightarrow e^{-\frac{r}{a}}} .

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\sqrt{\frac{-\hbar^2}{2m E}}}

If we allow Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \kappa=a^{-1}\!} , then the large limit of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r\!} can be expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)\sim e^{-\kappa r}\!}

Using the limits of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u(r)\!} , the wavefunction can be expressed as the following

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(r)=(\kappa r)^{l+1}e^{-\kappa r}W(\kappa r)\!}

To simplify the equation, make a substitution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho=\kappa r\!} . The equation now turns into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_l(\rho)=\rho^{l+1}e^{-\rho}W(\rho)\!}

Plugging this equation into the Schrödinger equation and simplifying, it turns into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d^2W(\rho)}{d\rho^2}+2\left(\frac{l+1}{\rho}-1\right)\frac{dW(\rho)}{d\rho}+\left(\frac{\rho_0}{\rho}-\frac{2(l+1)}{\rho}\right)W(\rho)=0} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho _{0}=Ze^{2}\sqrt {\frac{2\mu }{\hbar ^{2}E}} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\rho)\!} can be expressed as an expansion of polynomials;

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\rho)=a_0+a_1 \rho+a_2\rho^2+...=\sum_{k=0}^\infty a_k \rho^k}

The Schrödinger equation is then expressed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^\infty (a_{k}k(k-1)\rho^{k-2}+2(l+1)k\rho^{k-2}a_k-2\rho^{k-1}a_k k)+\sum_{k=0}^\infty(\rho_0 a_k\rho^{k-1}-2(l+1)a_k\rho^{k-1})=0}

And simplified into

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{k=0}^\infty (a_{k+1}(k+1)k\rho^{k-1}+2(l+1)(k+1)\rho^{k-1}a_{k+1}-2\rho^{k-1}a_k k+(\rho_0-2(l+1))a_k\rho^{k-1})=0}

Bring all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho\!} 's to the same power

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k(k+1)a_{k+1}+2(l+1)(k+1)a_{k+1}-2ka_k+(\rho_0-2(l+1))a_k=0\!}

which can be expressed in the simplest fractional form as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{a_{k+1}}{a_k}=\frac{2(k+l+1)-\rho_0}{(k+1)(k+2l+2)}}

where $\rho _{0}=2(N+l+1)\!$ and $N=0,1,2...\!$ and $l=0,1,2,...\!$

In the limit of large k

$\lim _{k\to \infty }{\frac {a_{k+1}}{a_{k}}}\rightarrow {\frac {2}{k}}$

$a_{k+1}={\frac {2}{k}}a_{k}\sim {\frac {2^{k}}{k!}}$

this will make

$\psi (r)\sim e^{kr}\rightarrow \infty$

so we need to break, and make

$a_{k+1}=0\!$

from this, we get the energy spectrum.

The fractional form can be expressed as a confluent hypergeometric function

$_{1}F_{1}(a,c,z)=1+{\frac {a}{c}}{\frac {z}{1!}}+{\frac {a(a+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+...=\sum _{k=0}^{\infty }a_{k}z^{k}$

It should be noted that a confluent hypergeometric function is a solution of Kummer's equation, which is:

$z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0.\,\!$

Derriving the recursion relations we have:

${\frac {a_{k}}{a_{k-1}}}={\frac {\frac {a(a+1)(a+2)...(a+k-1)}{c(c+1)(c+2)...(c+k-1)k!}}{\frac {a(a+1)(a+2)...(a+k-2)}{c(c+1)(c+2)...(c+k-2)(k-1)!}}}={\frac {a+k-1}{(c+k-1)k}}$

${\frac {a_{k+1}}{a_{k}}}={\frac {a+k}{(c+k)(k+1)}}$

by comparison, we find that

$c=2(l+1)\!$

$a=-N\!$

$z=2\rho \!$

So the solution of $W(\rho )\!$ is

$W(\rho )=constant_{1}F_{1}(-N,2(l+1),2\rho )\!$

where :$\rho =\kappa r={\sqrt {\frac {-2\mu E}{\hbar ^{2}}}}r$ Full wavefunction solution with normalization is

$\psi _{n,l,m}(r,\theta ,\phi )={\frac {e^{-\kappa r}(2\kappa r)^{l}}{(2l+1)!}}{\sqrt {\frac {(2\kappa )^{3}(n+l)!}{2n(n-l+1)}}}\ _{1}F_{1}(-n+l+1,2(l+1),2\kappa r)Y_{lm}(\theta ,\phi )$

The first couple of normalized wavefunctions for the hydrogen atom are as follows

$\psi _{100}={\dfrac {e^{-r/a_{o}}}{\sqrt {\pi a_{o}^{3}}}}$

$\psi _{200}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left(2-{\dfrac {r}{a_{o}}}\right)$

$\psi _{210}={\dfrac {e^{-r/2a_{o}}}{\sqrt {32\pi a_{o}^{3}}}}\left({\dfrac {r}{a_{o}}}\right)cos(\theta )$

$\psi _{2\pm 10}={\dfrac {e^{-r/2a_{o}}}{\sqrt {64\pi a_{o}^{3}}}}\left({\dfrac {r}{a_{o}}}\right)sin(\theta )e^{\pm i\phi }$

The energy is then found to be

$E=-{\frac {Z^{2}e^{4}\mu }{2\hbar ^{2}n^{2}}}=-{\frac {Z^{2}}{n^{2}}}Ry$

where $Ry=13.6eV\!$ for the hydrogen atom and $n=1,2,3,...\!$ and the degeneracy for each level is $n^{2}\!$.

To the side is a chart depicting the energy levels for hydrogen atom for $n=1,2,3\!$ in units of $Ry$. (The parenthesis indicates the degeneracy due to possibile values of the magnetic quantum number $m$ from $-l$ to $+l$).

Problems

Problem 1
Problem 2