Show that the probability of obtaining En for a particle in a box with the provided wave function (see first image) is given by the equation in the first image. Hint: The probability of obtaining the ground-state energy E1 is determined in Example 3.4, as shown in the second provided image. Show that the probability of obtaining E,, for aparticle in a box with wave function(x) =is given by3x(L − x) 0 < x < Lelsewhere{√ EXCL_|c₁|²=240поль[1-(-1)"]² EXAMPLE 3.4 The wave function for a particle in the box is given by30{√BX(L_4(x) =Example 2.5 shows that is normalized. A measurement of the energy of theparticle is carried out. What is the probability of obtaining E₁, the ground-stateenergy for the particle in a box?C1SOLUTION To determine the probability, evaluate c₁:= [₁ v₁(x)(x) dx = [₁ √ ² sinLx(L-x) 0

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Answered: Show that the probability of obtaining…

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Consider an electron in a one-dimensional, infinitely-deep, square potential well of width d. The electron is in the ground state. (a) Sketch the wavefunction for the electron. Clearly indicate the position of the walls of the potential well on your sketch. (b) Briefly explain how the probability distribution for detecting the electron at a given position differs from the wavefunction.
The normalised wavefunction for an electron in an infinite 1D potential well of length 80 pm can be written:ψ=(0.587 ψ2)+(0.277 i ψ7)+(g ψ6). As the individual wavefunctions are orthonormal, use your knowledge to work out |g|, and hence find the expectation value for the energy of the particle, in eV.
An Infinite Square Well of width L that is centred around x = 0 is shown in the figure. At t = 0, a particle exists in this system with the wavefunction provided, where Ψ0 is √(12/L), and Ψ = 0 for all other values of x. Calculate the probability density for this particle at t = 0, and state the position at which it takes its maximum value. then, calculate the expectation value for the position of this particle at t = 0, i.e. ⟨ x⟩. Compare the results of the positions found and explain why they are different.
Problem 15. In the case of a particle tunneling through the barrier, we made equal both functions and derivatives at the boundary between region I and II, and between II and III, see eq. (9a – 9d). But in the case of the particle in the box, we made both functions equal at x = 0 and at x = L. Explain why we did not consider derivatives in this case.
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1.The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ ? ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc a) Sketch the potential of this system , including in your sketch the positions of the lowest three energy levels . Indicate in your sketch the form of the wavefunction for a particle in each of these energy levels , and state which of the wavefunctions you have drawn could be decirbed by the Ψn written above (see figure) . b) Calculate the expectation value of momentum , ⟨p⟩ for a particle with n=2 c) Calculate the expectation value of momentum squared ⟨p 2⟩ , for a particle with n = 2 . Hint : you may use the mathematical identiy sin2 x = 1/2 (1 − cos 2x) without proof .

Show that the probability of obtaining En, for a particle in a box with wave function Y(x)= -{√B. is given by |c₁|² 30x (L-x) 0