A particle is confined in a box of length L as shown in the figure. If the potential is treated as a perturbation, including the first order correction, the ground state energy is ħ²π² Vo (a) E = + Vo (b) E: ħ²π² 2mL² 2mL² ħ² π² ħ² r² (c) E = (d) E= - 2mL² 2mL² + Vo 0 4 - + ON 2 L/2
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- A particle of mass 1.60 x 10-28 kg is confined to a one-dimensional box of length 1.90 x 10-10 m. For n = 1, answer the following. (a) What is the wavelength (in m) of the wave function for the particle? m (b) What is its ground-state energy (in eV)? eV (c) What If? Suppose there is a second box. What would be the length L (in m) for this box if the energy for a particle in the n = 5 state of this box is the same as the ground-state energy found for the first box in part (b)? m (d) What would be the wavelength (in m) of the wave function for the particle in that case? mQuestion A6 Consider an infinite square well with V = 0 in the interval -L/2 < x < L/2, and V → ∞ everywhere else. A particle of mass m is in the groundstate of this system, and is known to have a wavefunction and energy given by TX √ = COS and E = π²h² 2mL² The system is then perturbed so that its potential takes the constant value VConsider the 1D time-independent Schrodinger equation ħ² ď² 2m dr² with the potential where to is a parameter. (a) Show that V(x) = +V(x)] = Ev v is a solution of the Schrodinger equation. ħ² mx² sech² 1 = A sech x xo (₁)The normalised wavefunction for an electron in an infinite 1D potential well of length 89 pm can be written:ψ=(-0.696 ψ2)+(0.245 i ψ9)+(g ψ4). If the state is measured, there are three possible results (i.e. it is in the n=2, 9 or 4 state). What is the probability (in %) that it is in the n=4 state?a question of quantum mechanics: Consider a particle in a two-dimensional potential as shown in the picture Suppose the particle is in the ground state. If we measure the position of the particle, what isthe probability of detecting it in region 0<=x,y<=L/2 ?3