Consider a classical particle of mass M M M F = − M ω 2 x F = -M\omega^2 x F = − M ω 2 x ω \omega ω k = M ω 2 k = M\omega^2 k = M ω 2
Write down the potential energy of the particle, as a function of its position x x x
When the particle is at maximum displacement, what is its kinetic energy?
Use your answers to parts 1. and 2. to derive an equation which gives the amplitude of oscillation as a function of the energy of the particle.
Assume now that the energy of the particle is equal to the energy of the n t h n^\mathrm{th} n th quantum harmonic oscillator, E n = ℏ ω ( n + 1 2 ) E_n = \hbar \omega (n + \frac{1}{2}) E n = ℏ ω ( n + 2 1 )
Express the amplitude of the particle in terms of the characteristic length ℓ = ℏ M ω \ell = \sqrt{\frac{\hbar}{M\omega}} ℓ = M ω ℏ
Consider now a quantum particle, confined within a harmonic oscillator potential well (with the same ω \omega ω ∣ E 0 ⟩ \ket{E_0} ∣ E 0 ⟩
What does it mean for a particle to be tunnelling in quantum mechanics?
Use the previous parts of this question to determine the regions of space where the particle would be tunneling.
Calculate the total probability that, upon measurement of the position of the particle, it will be within the tunnelling region (you will need to evalute the integral numerically ).
What is the general expression for the position of a particle at time t t t
Use part 8. to determine the kinetic energy of the particle when it is at the origin.
Confirm that you can use the kinetic energy at the origin to arrive at the same expression relating the amplitude of oscillation and the energy.
Repeat the calculation in part 7, except now assume that the particle is in a different energy eigenstate ∣ E k ⟩ \ket{E_k} ∣ E k ⟩ k = 1 k = 1 k = 1