Quantum Chemistry/Example 4

Example 4Edit

Write a question about calculating the frequency of a photon to calculate the energy to transition between two levels of an electron in a 1D box

QuestionEdit

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 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 x 10-33 J. Since  , 3.01 x 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 x 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 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 x 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,