Quantum Chemistry/Example 4
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?
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 J s), 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.
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 J s), 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,