Question

The Hamiltonian \(H\) for the hydrogen atom is $$ -\frac{\hbar^{2}}{2 m} \nabla^{2}-\frac{q}{4 \pi \epsilon_{0} r} $$ For a spherically symmetric state, as may be assumed for the ground state, the only relevant part of \(\nabla^{2}\) is that involving differentiation with respect to \(r\). (a) Define the integrals \(J_{n}\) by $$ J_{n}=\int_{0}^{\infty} r^{n} e^{-2 \beta r} d r $$

Short answer

The integral \( J_{n} \) can be defined recursively as \( J_{n} = \frac{n}{2 \beta} J_{n-1} \).

Step-by-step solution

Title - Rewrite the integral

First, rewrite the integral definition of \(J_{n}\) as given in the problem statement:\[ J_{n} = \int_{0}^{\infty} r^{n} e^{-2 \beta r} \, dr \]

Title - Use integration by parts

Apply integration by parts to solve the integral. Integration by parts is given by the formula:\[ \int u \, dv = uv - \int v \, du \]For our integral, let:\[ u = r^{n} \quad \text{and} \quad dv = e^{-2 \beta r} \, dr \]Then calculate \(du\) and \(v\):\[ du = n r^{n-1} \, dr \]\[ v = \frac{e^{-2 \beta r}}{-2 \beta} \]

Title - Calculate the integral using parts

Now apply the integration by parts formula:\[ J_{n} = \left. r^{n} \cdot \left( \frac{e^{-2 \beta r}}{-2 \beta} \right) \right|_{0}^{\infty} - \int_{0}^{\infty} \left( \frac{e^{-2 \beta r}}{-2 \beta} \right) \cdot n r^{n-1} \, dr \]

Title - Evaluate the boundaries

Evaluate the boundaries at 0 and \(\infty\):\[ \left. \frac{r^{n} e^{-2 \beta r}}{-2 \beta} \right|_{0}^{\infty} = 0 - 0 = 0 \]This simplifies to:\[ J_{n} = \int_{0}^{\infty} \frac{n r^{n-1} e^{-2 \beta r}}{2 \beta} \, dr \]

Title - Recognize recursive nature

Recognize that the remaining integral is a scaled version of our original integral \( J_{n} \). Specifically, it becomes:\[ J_{n} = \frac{n}{2 \beta} J_{n-1} \]

Title - Solve the recursion

To find \( J_{n} \), solve the recursive relation step-by-step starting from an initial condition like \( J_{0} \). \( J_{0} \) can be solved directly and then use to solve \( J_{1} \), \( J_{2} \), and so on.

Key concepts

Integration by Parts

Integration by parts is a fundamental technique in calculus used to transform the integral of a product of functions into simpler components. It's formulated as: \[ \int u \, dv = uv - \int v \, du \] In this problem, we applied integration by parts to the integral \[ J_{n} = \int_{0}^{\infty} r^{n} e^{-2 \beta r} \, dr \]. We set:

• u = r^n
• dv = e^{-2 \beta r} \, dr

By computing the derivatives and integrals, we found:

• du = n r^{n-1} \, dr
• v = \frac{e^{-2 \beta r}}{-2 \beta}

Using these components, we manipulate our integral effectively, which helps us simplify complex integral problems like this one.

Spherical Symmetry

Spherical symmetry is a key concept in many areas of physics, particularly in the field of quantum mechanics for systems like the hydrogen atom. In a spherically symmetric system, physical properties do not change with rotation. For the hydrogen atom, this symmetry simplifies our analysis because we can ignore angular dependence and consider only radial dependence.
For the given Hamiltonian, \[H = -\frac{\hbar^{2}}{2 m} abla^{2} - \frac{q}{4 \ pi \epsilon_{0} r} \], the Laplacian \(abla^{2}\) focuses solely on the radial part when assuming spherical symmetry. This dramatically simplifies our integrals and the overall solution, allowing us to work with one-dimensional integrals in the radial coordinate \(r\).

Recursion in Integrals

Recursion in integrals allows us to solve complex integrals by relating them to simpler ones. In our exercise, after applying integration by parts, we derived a relation between \( J_{n} \) and \( J_{n-1} \): \[J_{n} = \frac{n}{2 \beta} J_{n-1} \]. This recursive formula means that if we know the value of \( J_{n-1} \), we can compute \( J_{n} \).

To completely solve for \( J_{n} \), we need an initial condition like \( J_{0} \), which we can often compute directly. By recursively applying our formula, we can build up from smaller values to larger ones, making the process manageable and systematic.

Quantum Mechanics

Quantum mechanics is the branch of physics that studies phenomena at microscopic scales, around the size of atoms and subatomic particles. The hydrogen atom, being one of the simplest systems, is a classic subject in quantum mechanics.
The Hamiltonian defined in our problem describes the energy of a particle (a single electron in the hydrogen atom) considering both kinetic energy (represented by the Laplacian \( abla^{2} \)) and potential energy (due to electrostatic attraction between the electron and proton, given by \( -\frac{q}{4\pi \epsilon_{0} r} \)).
In spherical coordinates, the Schrodinger equation simplifies considering the spherical symmetry, making problems more tractable and allowing in-depth study of electron behavior in atoms.