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consider a particle of mass 10-30 kg trapped in an infinitely deep potential well of width 0.59 nm. the value of k in ground state is approximately
- a)2.3nm
- b)6.3nm-1
- c)2.3nm-1
- d)2.3
Correct answer is option 'B'. Can you explain this answer?
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consider a particle of mass 10-30 kg trapped in an infinitely deep pot...
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consider a particle of mass 10-30 kg trapped in an infinitely deep pot...
Particle in an infinitely deep potential well
The problem describes a particle of mass 10-30 kg trapped in an infinitely deep potential well of width 0.59 nm. We are asked to find the value of k in the ground state.
Understanding the problem
In quantum mechanics, a particle trapped in a potential well is described by a wave function. The wave function represents the probability amplitude of finding the particle at a given position. The width of the potential well determines the allowed energy levels of the particle.
Calculating the value of k
To find the value of k in the ground state, we can use the formula for the energy levels of a particle in an infinitely deep potential well:
E = (n^2 * h^2) / (8 * m * L^2)
where E is the energy, n is the quantum number (1, 2, 3, ...), h is the Planck's constant, m is the mass of the particle, and L is the width of the potential well.
In the ground state, n = 1. Plugging in the given values:
E = (1^2 * h^2) / (8 * m * L^2)
We are given the mass of the particle as 10-30 kg and the width of the potential well as 0.59 nm.
E = (h^2) / (8 * 10^-30 kg * (0.59 * 10^-9 m)^2)
Simplifying the expression:
E = (h^2) / (8 * 10^-30 * 0.59^2 * 10^-18)
E = (h^2) / (8 * 5.481 * 10^-48)
E = (6.626 * 10^-34 J*s)^2 / (8 * 5.481 * 10^-48)
E = 2.3 * 10^-19 J
Since k is related to the energy by k = sqrt(2mE) / h, we can calculate the value of k:
k = sqrt(2 * 10^-30 kg * 2.3 * 10^-19 J) / (6.626 * 10^-34 J*s)
k ≈ 6.3 nm^-1
Therefore, the correct answer is option 'B': 6.3 nm^-1.
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