# Quantum Chemistry/Example 4

## Question

[edit | edit source]An electron in a 1D box emits a photon as the electron transitions to a lower energy level. If the length of the 1D box is equal to 1.0 cm, and the quantum number transition is , what is the electromagnetic radiation frequency of the emitted photon?

**Solution:**

The energy level of a particle in a 1D box at a specific quantum number () is,

Where is equal to Planck's constant (6.62607015 x 10^{-34} Js), is equal to the quantum number ( = 1, 2, 3, ...), is equal to the mass of the particle, and is equal to the length of the 1D box. For an electron, the mass is equal to 9.10938356 x 10^{-31} kg.

Since (assuming the mass and box length are constant), the energy level increases by a factor of 4 as the quantum number increases by a factor of 2. Therefore, if a particle in a 1D box undergoes an energy level transition, there is a difference between the initial and final quantum number energy levels. The energy level difference () of a particle in a 1D box that has undergone an energy level transition is,

Where is equal to the final quantum number, and is equal to the initial quantum number.

If , is a positive value; photon absorbed.

If , is a negative value; photon emitted.

Therefore, the energy level difference of an electron in a 1D box with a length of 1.0 cm, which has undergone a transition is,

The energy level difference for the electron which underwent a transition is equal to -3.01 × 10^{-33} J. Since , 3.01 × 10^{-33} J was emitted from the electron. If the electron underwent a transition, the electron would absorb the same amount of energy that was emitted from the transition which was 3.01 × 10^{-33} J.

Therefore,

The energy of a photon has a specific frequency of electromagnetic (EM) radiation, and the energy is directly proportional to the frequency. The energy of the photon is equal to,

Where is equal to Planck's constant (6.62607015 x 10^{-34} Js), and is equal to EM radiation frequency.

Rearranging this equation allows for the calculation of the photon EM radiation frequency,

The calculated photon energy was equal to 3.01 × 10^{-33} J, therefore the EM radiation frequency of the emitted photon from the transition of an electron in a 1D box with a length of 1.0 cm is,