Quantum Energy Levels Calculator: Particle in a Box, Hydrogen Atom and Oscillator

Allowed energies of a particle in a box, a hydrogen atom and a quantum oscillator, and the photon from a jump.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Choose the system. For a particle in an infinite box enter the width and the particle: the energies are n squared times h squared over 8 m L squared. For hydrogen the energies are minus 13.6 electron volts over n squared. For a harmonic oscillator they are h f times n plus a half.

Enter a starting and a final level and it gives the energy of the photon given out or taken in, its frequency and its wavelength.

These are the solutions of the time-independent Schrodinger equation H psi = E psi, an eigenvalue equation where the Hamiltonian H is the kinetic plus the potential energy. The energies are quantised, so only certain values are allowed.

Formulas used

  • E_n = \frac{n^2h^2}{8mL^2}
  • E_n = -\frac{13.6\,\mathrm{eV}}{n^2}
  • E_n = h\nu\left(n + \tfrac{1}{2}\right)
  • \lambda = \frac{hc}{\Delta E}

Worked example

With the default numbers in the calculator above:

  1. Energy of the first level (n = 1): 0.37603 eV
  2. Energy of level 2: 1.5041 eV
  3. Energy of level 1: 0.37603 eV
  4. Photon given out in the jump: energy: 1.1281 eV
  5. Photon frequency: 272.77 THz
  6. Photon wavelength: 1099.1 nm

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Which system? Hydrogen atom; Starting level n (higher for emission; the oscillator starts at 0, the box and hydrogen at 1) 3; Final level n 2.

  1. Energy of the ground state (n = 1): -13.606 eV
  2. Energy of level 3: -1.5117 eV
  3. Energy of level 2: -3.4014 eV
  4. Photon given out in the jump: energy: 1.8897 eV
  5. Photon frequency: 456.92 THz
  6. Photon wavelength: 656.11 nm

Common questions

Why is the lowest energy not zero?

The uncertainty principle: a particle confined in a small space must have some momentum, so some energy.

Which hydrogen lines are visible?

The Balmer series, jumps down to level 2.

What is the Hamiltonian?

The operator for the total energy of the system: the kinetic energy plus the potential energy.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Griffiths, Introduction to Quantum Mechanics; CODATA 2018 physical constants.

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