Question

A particle in an infinite well (U = 0, when 0 ≤ x ≤ L; U = [infinity], when x < 0 or x > L) is in the ground state (n=1) with an energy of 1.26 eV. How much energy must be added to the particle to reach the second excited state (n=3)?

Answer 1

Step-by-step explanation:

In an infinite well potential, the energy levels are given by the formula:

E_n = (n^2 * h^2) / (8 * m * L^2)

where E_n is the energy of the nth state, n is the quantum number, h is the Planck's constant, m is the mass of the particle, and L is the length of the well.

We are given that the particle is in the ground state (n=1) with an energy of 1.26 eV. Let's convert this energy to joules:

1 eV = 1.6 x 10^-19 J

So, the energy of the ground state is:

E_1 = 1.26 eV * (1.6 x 10^-19 J/eV)

E_1 = 2.016 x 10^-19 J

To find the energy required to reach the second excited state (n=3), we need to calculate the energy of the third state (E_3) using the same formula:

E_3 = (3^2 * h^2) / (8 * m * L^2)

The energy to be added is the difference between E_3 and E_1:

Energy required = E_3 - E_1

Substituting the values, we have:

Energy required = [(3^2 * h^2) / (8 * m * L^2)] - 2.016 x 10^-19 J

Please note that the values of h, m, and L are required to calculate the exact energy difference.

Answer 2

A particle initially in the ground state with an energy of 1.26 eV requires an addition of 10.08 eV to reach the second excited state (n=3) in an infinite square well.

To calculate the energy that must be added to a particle to reach the second excited state in an infinite square well, we need to understand the energy levels of a quantum particle in such a well.

The ground state energy corresponds to n=1, and for an infinite well, the energy levels are given by the relationship En = n²E1, where,

E1 is the ground state energy and

n is the quantum number corresponding to the energy level.

Given that the ground state energy is 1.26 eV, the energy of the second excited state (n=3) will be E3 = 3² × 1.26 eV = 9 × 1.26 eV = 11.34 eV.

To determine how much energy must be added to reach this state from the ground state, we subtract the ground state energy from the second excited state energy : 11.34 eV - 1.26 eV = 10.08 eV.

Therefore, to move a particle from the ground state to the second excited state in an infinite square well, an energy of 10.08 eV must be added to the particle.