How to Calculate the Ground State of a Hydrogen Atom
The hydrogen atom is composed of two particles: an electron and a proton. To calculate the ground state of the hydrogen atom, you must solve Schrodinger's equation for the interaction of a proton and electron. You can do most of the calculation by hand if you are starting from scratch, and, if you are good with numbers, you may not need a calculator at all.
Instructions
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1
Review basic quantum theory and mathematical tools such as vector calculus. Without a good foundation in these subjects, you'll have an extremely difficult time understanding the calculation.
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2
Realize that the Coulomb force between an electron and proton is much stronger than the gravitational force between them. As a result, you can ignore the gravitational force and deal only with the Coulomb (or electric) force.
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3
Write the time-independent Schrodinger equation for the hydrogen atom. Written concisely and generally, the Schrodinger equation is H Psi = E Psi, where H is the Hamiltonian operator, E is an energy value and Psi is the wave function. The Hamiltonian for the hydrogen atom is (-hbar^2 / 2m * Del^2 + U), where hbar is Planck's constant divided by 2 Pi, m is the reduced mass of the atom, Del^2 is the Laplacian operator and U is the Coulomb potential.
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4
Use separation of variables on the wave function, Psi, to get three equations, each involving only a single variable. Since the full Schrodinger equation in this case has three variables (corresponding to the spatial dimensions), you must separate this multivariable equation into three single-variable equations. Your best choice here is to use spherical coordinates.
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5
Solve each of the three equations obtained in Step 4. You will need to introduce three so-called quantum numbers, which you can label n, l and m.
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Recombine the solutions to each equation to get the wave function of the hydrogen atom. Simply multiply the three solutions from Step 5.
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7
Find the ground state (and corresponding lowest energy level) of the hydrogen atom by setting n equal to 1 and l and m equal to zero. This gives you both the wave function and the energy of the atom.
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Tips & Warnings
Calculating the energy levels of the hydrogen atom is a subject that could take an entire course (or two) at a university. These steps provide only an overview of the procedure; they don't explain all the mathematical details and physical concepts.
