Question

Find the energy eigenvalues and eigenfunctions for the hydrogen atom. The potential energy is \(V(r)=-e^{2} / r\) in Gaussian units, where \(e\) is the charge of the electron and \(r\) is in spherical coordinates. since \(V\) is a function of \(r\) only, you know from Problem 18 that the eigenfunctions are \(R(r)\) times the spherical harmonics \(Y_{l}^{m}(\theta, \phi),\) so you only have to find \(R(r) .\) Substitute \(V(r)\) into the \(R\) equation in Problem 18 and make the following simplifications: Let \(x=2 r / \alpha, y=r R ;\) show that then $$r=\alpha x / 2, \quad R(r)=\frac{2}{\alpha x} y(x), \quad \frac{d}{d r}=\frac{2}{\alpha} \frac{d}{d x}, \quad \frac{d}{d r}\left(r^{2} \frac{d R}{d r}\right)=\frac{2}{\alpha} x y^{\prime \prime}$$ Let \(\alpha^{2}=-2 M E / \hbar^{2}\) (note that for a bound state, \(E\) is negative, so \(\alpha^{2}\) is positive) and \(\lambda=M e^{2} \alpha / \hbar^{2},\) to get the first equation in Problem 22.26 of Chapter \(12 .\) Do this problem to find \(y(x),\) and the result that \(\lambda\) is an integer, say \(n\). [Caution: not the same \(n \text { as in equation }(22.26)] .\) Hence find the possible values of \(\alpha\) (these are the radii of the Bohr orbits), and the energy eigenvalues. You should have found \(\alpha\) proportional to \(n ;\) let \(\alpha=n a,\) where \(a\) is the value of \(\alpha\) when \(n=1,\) that is, the radius of the first Bohr orbit. Write the solutions \(R(r)\) by substituting back \(y=r R\) and \(x=2 r /(n a),\) and find \(E_{n}\) from \(\alpha\).

Short answer

The energy eigenvalues are given by \[ E_n = -\frac{Me^4}{2\hbar^2 n^2} \].

Step-by-step solution

Understand the given potential energy

The potential energy for the hydrogen atom is given by the function: \[ V(r) = -\frac{e^2}{r} \]where \(e\) is the charge of the electron and \(r\) is the radial distance.

Identify the form of the eigenfunctions

The eigenfunctions are of the form: \[ R(r) Y_l^m(\theta, \phi) \]with \(R(r)\) being the radial part and \(Y_l^m(\theta, \phi)\) the spherical harmonics. We need to solve for \(R(r)\).

Convert variables and expressions

Using the substitutions: \[ x = \frac{2r}{\alpha}, y = rR \] we obtain: \[ r = \frac{\alpha x}{2}, \quad R(r) = \frac{2}{\alpha x} y(x) \]Therefore: \[ \frac{d}{dr} = \frac{2}{\alpha} \frac{d}{dx}, \quad \frac{d}{dr}\left(r^2 \frac{dR}{dr}\right) = \frac{2}{\alpha} x y'' \]

Define constants \( \alpha \) and \( \lambda \)

Let: \[ \alpha^2 = -\frac{2ME}{\hbar^2} \quad \text{and} \quad \lambda = \frac{Me^2\alpha}{\hbar^2} \]

Solve for \( y(x) \)

Substitute the expressions into the radial equation, and show that \( \lambda \) needs to be an integer, say \( n \).

Determine possible values of \( \alpha \)

Since \( \alpha \) is proportional to \( n \), let \( \alpha = na \), where \( a \) is for \( n = 1 \).

Find energy eigenvalues

Using \( \alpha = na \), solve for energy eigenvalues: \[ E_n = - \frac{Me^4}{2\hbar^2 n^2} \]

Write the solutions \( R(r) \)

Substitute back \( y = rR \) and \( x = \frac{2r}{na} \) to express: \[ R(r) = \frac{2}{\alpha x} y(x) \]

Key concepts

Schrödinger Equation

The Schrödinger equation is the fundamental equation of quantum mechanics. It's used to describe how the quantum state of a physical system changes over time. For the hydrogen atom, we set up the time-independent Schrödinger equation because we're looking for energy eigenvalues. This equation can be written as:

\[ -\frac{\bar{h}^2}{2m} abla^2 \text{\tiny +} V(r) \text{\tiny } \text{\tiny } \text{\tiny } \text{\tiny } \text{\tiny } \text{\tiny } \text{(r,\theta,\bbphi)} \] \[ \text{\tiny +} E \text{\tiny (r)} \] Equation applications often require solving differential equations in spherical coordinates for atoms like hydrogen.

Quantum Mechanics

Quantum mechanics is a branch of physics that deals with phenomena at very tiny scales, like atoms and subatomic particles. It's essential for understanding the behavior of electrons within an atom. Principles of quantum mechanics enable us to calculate quantities such as energy levels and state functions of the hydrogen atom by solving the Schrödinger equation. The energy eigenvalues and corresponding eigenfunctions help define the quantized states that electrons can occupy within an atom.

Spherical Harmonics

Spherical harmonics are special functions defined on the surface of a sphere. They appear in the solutions to the Schrödinger equation in spherical coordinates. They are denoted as \( Y_l^m(\theta, \bbphi) \), where \( l \) and \( m \) are quantum numbers.

The role of spherical harmonics in the hydrogen atom problem is to describe the angular part of the wavefunction. The wavefunction of the electron can thus be separated into radial and angular parts, with spherical harmonics helping to solve the angular part.

Bohr Radius

The Bohr radius \( a_0 \) is a physical constant that characterizes the size of the hydrogen atom. It is the approximate distance between the proton and the electron in a ground state hydrogen atom:

\[ a_0 = \frac{4 \bbpi \bbvarepsilon_0 \bbhbar^2}{m_e e^2} \]

In the context of our problem, the Bohr radius helps us define the scale for the radial coordinate. It appears naturally when solving the radial part of the Schrödinger equation and determining the energy eigenvalues.

Eigenfunctions

Eigenfunctions are special solutions to the Schrödinger equation that correspond to specific energy eigenvalues \( E_n \). For the hydrogen atom, the eigenfunctions are described by the radial function \( R(r) \) and spherical harmonics \( Y_l^m(\theta, \bbphi) \).

The eigenfunctions give the probability distribution of the electron's location around the nucleus. By solving the Schrödinger equation, we determine these eigenfunctions and the allowed energy states that electrons can occupy.

Potential Energy

The potential energy \( V(r) \) in a hydrogen atom due to the electrostatic attraction between the positively charged proton and negatively charged electron is defined as:

\[ V(r) = -\frac{e^2}{r} \]

This potential energy is a function of the radial distance \( r \). It plays a crucial role in the Schrödinger equation, influencing the form of the radial equation and the solutions we obtain for the energy eigenvalues and eigenfunctions.

Radial Equation

The radial equation comes from the separation of variables in the Schrödinger equation for spherical coordinates. This separates the wavefunction into a radial part \( R(r) \) and an angular part described by spherical harmonics. The radial part must satisfy:

\[ -\frac{\bar{h}^2}{2m} \frac{d^2R}{dr^2} + V(r)R(r) = ER(r) \]

Solving the radial equation involves finding \( R(r) \) through a series of variable transformations and defining the constants \( \alpha \) and \( \lambda \). The solutions yield the hydrogen atom's energy eigenvalues and the radial part of its eigenfunctions.