Exercises : Questions

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1 Exercises : Questions Problem 1 where Calculate the following commutators: a) [ Ĥ, ˆp ], b) [ Ĥ, ˆr ], Ĥ = 1 2m ˆp2 + V ˆr), 1) ˆp 2 = ˆp 2 x + ˆp 2 y + ˆp 2 z and V ˆr) = V ˆx, ŷ, ẑ) is an arbitrary good function. Hint 1 First solve the problem 3 of the file Exercises Problem 2 E, Let Ĥ be the Hamiltonian of a system. Suppose that Ĥ has a degenerate eigenvalue Ĥψ α = Eψ α, α = 1, 2,..., D), 2) where D is the degree of degeneracy of the energetic level E, that is D is the dimensionality of the subspace spanned by {ψ α } and corresponding to the eigenvalue E. Let D ϕ β = a βα ψ α, β = 1, 2,..., D), 3) α=1 be an arbitrary linear combination of the eigenfunctions ψ α belonging to the same degenerate subspace of dimension D. It is evident that the functions ϕ β are still eigenfunctions of Ĥ with the same eigenvalue E, namely D ) Ĥϕ β = a βα Ĥψα = E α=1 D a βα ψ α α=1 = Eϕ β. 4)

2 Prove that it is possible to choose the set of new eigenfunctions {ϕ β } such that they are orthonormal, namely ϕ αq)ϕ β q) dq = δ αβ, 5) and that this can be done in infinitely many different ways. Hint 2 Calculate the number of independent coefficients of the linear transformation ψ α ϕ β, and the number of constraints implied by the orthonormality condition 5). Problem 3 The probability density and the probability current jr, t) satisfy the continuity equation Calculate ρr, t) and jr, t) for where ψr, t) = A exp ρr, t) ψr, t) 2, 6) [ ψ r, t) ψr, t) ψr, t) ψ r, t) ], 7) 2m i ρ t + j = 0. 8) i r p ) iet + B exp i r p ) iet, 9) E = p p 2m, 10) and A, B C. Verify that ψr, t) satisfy Schrödinger s equation, check that ρr, t) and jr, t) fulfill the continuity equation 8) and discuss the results. Solution 3 The wave function 9) is factorable in the product of a spatially-dependent function fr) and a time-dependent function T t), that is [ ψr, t) = A exp i r p ) + B exp i r p exp iet/) fr)t t). 11) 2

3 Since T t) 2 = 1, 12) then the probability density ρr, t) = ψr, t) 2 = fr) 2 is automatically time independent and, therefore, ρ t = 0. 13) It is easy to see that ψr, t) satisfies the Schrödinger equation for a free particle, because i 2 ψr, t) = t 2m 2 ψr, t), 14) i t ψr, t) = ifr) t T t) = ifr) [ i E T t) ] = Eψr, t), 15) and and ψr, t) = T t) fr) = ip [ A exp i r p ) B exp i r p T t), 16) Therefore 14) becomes 2 ψr, t) = [ ψr, t = p p [ A exp i r p ) 2 + B exp i r p T t) = p p ψr, t). 17) 2 [ Eψr, t) = 2 p p ] 2m ψr, t) 2 which is indeed an equality if E = p p/2m). The probability current is also time-independent because = p p ψr, t), 18) 2m jr, t) = [ ψ r, t) ψr, t) ψr, t) ψ r, t) ] 2m i = [ f r) fr) fr) f r) ]. 19) 2m i 3

4 Then, using 16) is easy to see that [ f r) fr) = A exp i r p ) ip [ A exp i r p ) = ip + B exp B exp [ A 2 B 2 + AB exp i r p i r p 2i r p ) A B exp and that fr) f r) = ip [ A 2 B 2 + A B exp 2i r p ) AB exp 2i r p 2i r p, 20). 21) Summing the last two equations and using 19), we obtain jr, t) = [ ip 2m i 2 A 2 B 2) ] = p A 2 B 2). 22) m Clearly, from 22) it follows that jr, t) is both time-independent and space-independent. Therefore, j = 0 and the continuity equation is satisfied because we already know that ρ/ t = 0. It should be noticed that ψr, t) is constructed as the sum of two counterpropagating waves with amplitudes A and B. When A = B the two waves generates a stationary wave and j = 0. Problem 4 Consider a classical particle in a one-dimensional box, moving in the region 0 < x < a with constant speed v. Imagine to observe the motion of the particle during the time interval t 0 t t 1 = t 0 + 2a/v, with t 0 arbitrary. Calculate the probability to find the particle between x and x+dx at time t, randomly peaked in the interval [t 0, t 1 ]. Compare your result with the quantum mechanical ones. Hint 4 Plot the diagram position-vs-time to visualize the trajectory of the particle in the interval [t 0, t 1 ]. Assume that t is a value assumed by the random variable τ uniformly distributed in the interval [t 0, t 1 ]. Learn how to calculate the probability density function p.d.f) of a given function of a uniformly distributed random variable. Problem 5 4

5 Consider a classical particle of mass m = 0.1 g, moving at a constant speed v = 1 m/s in a one-dimensional box of width a = 1 mm. Let E denotes the energy of this particle. Calculate the value of the index n such that the difference E E n = E 2 π 2 n2 2ma 2, 23) is minimum. Hint 5 Calculate the kinetic energy of the classical particle. Problem 6 This problem is given in Gasiorowicz s book, Example 3-5. Consider a particle in a box of width a. Its wave function is given by The probability density { A x a ψx) =, 0 < x < a/2, A ) 24) 1 x a, a/2 < x < a, where A is the normalization constant. Calculate A and the probability that a measurement of the energy yields the eigenvalue E n. Problem 7 This problem is given in Griffiths s book, Problem 2.8. A particle in the infinite square well the box) has the initial wave function where A > 0 is the normalization constant. ψx, 0) = Axa x), 25) a) Normalize ψx, 0). Graph it. Which stationary state does it most closely resemble? On that basis, guess the expectation value of the energy. b) Compute ˆx, ˆp, and Ĥ, at t = 0. Note: This time you cannot get ˆp by differentiating ˆx, because you only know ˆx at one instant of time.) How does Ĥ compare with your estimate in a)? c) Expand ψx, 0) in terms of the eigenfunctions of Ĥ. Plot the absolute square of the expansion coefficients as function of n. 5

6 Problem 8 A solution of the second-order ordinary differential equation with k > 0, is d 2 u dx 2 + k2 u = 0, 26) ux) = Ae ikx + Be ikx. 27) Given the boundary conditions for the particle in a box problem, u0) = 0, and ua) = 0, 28) which determines uniquely the constants A and B, prove that there are not other independent solutions. Hint 8 The linear independence of the solutions can be quantified calculating the Wronskian determinant. 6