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an electron is bound in 1D infinite well of width (1x10^-10)m . find the energy values in the ground state and first 2 excited state

Question

an electron is bound in 1D infinite well of width (1x10^-10)m . find the energy values in the ground state and first 2 excited state

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Solution

The energy levels of an electron in a 1D infinite potential well (also known as a particle in a box) can be calculated using the formula:

E_n = n²h² / 8mL²

where:

  • E_n is the energy of the level n
  • n is the quantum number (n=1 for the ground state, n=2 for the first excited state, n=3 for the second excited state, etc.)
  • h is the reduced Planck's constant (h = 6.62607015 × 10^-34 m² kg / s)
  • m is the mass of the electron (m = 9.10938356 × 10^-31 kilograms)
  • L is the width of the well

Let's calculate the energy for the ground state (n=1) and the first two excited states (n=2 and n=3).

  1. Ground state (n=1):

E_1 = 1² * (6.62607015 × 10^-34 m² kg / s)² / 8 * (9.10938356 × 10^-31 kilograms) * (1x10^-10 m)² E_1 = 6.0246836994 x 10^-19 Joules

  1. First excited state (n=2):

E_2 = 2² * (6.62607015 × 10^-34 m² kg / s)² / 8 * (9.10938356 × 10^-31 kilograms) * (1x10^-10 m)² E_2 = 2.40987347976 x 10^-18 Joules

  1. Second excited state (n=3):

E_3 = 3² * (6.62607015 × 10^-34 m² kg / s)² / 8 * (9.10938356 × 10^-31 kilograms) * (1x10^-10 m)² E_3 = 5.4222168292 x 10^-18 Joules

So, the energy of the ground state is approximately 6.02 x 10^-19 Joules, the energy of the first excited state is approximately 2.41 x 10^-18 Joules, and the energy of the second excited state is approximately 5.42 x 10^-18 Joules.

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