As a 1-dimensional problem, you are given a particle of mass, m, confined to a box of width, L. The initial wavefunction is given to you as: 4(x, t = 0) = C(x – L)x for 0
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- At time t = 0 a particle is described by the one-dimensional wave function 1/4 (a,0) = (²ª) e-ikre-ar² where k and a are real positive constants. Verify that the wave function (r, 0) is normalised. Hint: you may find the following standard integral useful: Loze -2² dx = √,A particle with mass m is in the state mx +iat 2h V (x, t) = Ae where A and a are positive real constants. Calculate the expectation value of (p).Consider a finite potential step with V = V0 in the region x < 0, and V = 0 in the region x > 0 (image). For particles with energy E > V0, and coming into the system from the left, what would be the wavefunction used to describe the “transmitted” particles and the wavefunction used to describe the “reflected” particles?
- Consider the one-dimensional step-potential V (x) = {0 , x < 0; V0 , x > 0}(a) Calculate the probability R that an incoming particle propagating from the x < 0 region to the right will reflect from the step.(b) Calculate the probability T that the particle will be transmitted across the step.(c) Discuss the dependence of R and T on the energy E of the particle, and show that always R+T = 1.[Hints: Use the expression J = (-i*hbar / 2m)*(ψ*(x)ψ′(x) − ψ*'(x)ψ(x)) for the particle current to define current carried by the incoming wave Ji, reflected wave Jr, and transmitted wave Jt across the step.For a simple plane wave ψ(x) = eikx, the current J = hbar*k/m = p/m = v equals the classical particle velocity v. The reflection probability is R = |Jr/Ji|, and the transmission probability is T = |Jt/Ji|. You need to write and solve the Schrodinger equation in regions x < 0 and x > 0 separately, and connect the solutions via boundary conditions at x = 0 (ψ(x) and ψ′(x) must be…consider an infinite square well with sides at x= -L/2 and x = L/2 (centered at the origin). Then the potential energy is 0 for [x] L/2 Let E be the total energy of the particle. =0 (a) Solve the one-dimensional time-independent Schrodinger equation to find y(x) in each region. (b) Apply the boundary condition that must be continuous. (c) Apply the normalization condition. (d) Find the allowed values of E. (e) Sketch w(x) for the three lowest energy states. (f) Compare your results for (d) and (e) to the infinite square well (with sides at x=0 and x=L)A Quantum Harmonic Oscillator, with potential energy V(x) = ½ mω02x2, where m is the mass of the particle in the potential, and ω0 is a constant. Determine the value of the quantum number n for the wavefunction provided. Explain how the result is obtained, as well as state a numerical value.
- Prove in the canonical ensemble that, as T ! 0, the microstate probability ℘m approaches a constant for any ground state m with lowest energy E0 but is otherwise zero for Em > E0 . What is the constant?Consider a particle moving in a 2D infinite rectangular well defined by V = 0 for 0 < x < L₁ and 0 ≤ y ≤ L2, and V = ∞ elsewhere. Outside the well, the wavefunction (x, y) is zero. Inside the well, the wavefunction (x, y) obeys the standing wave condition in the x and y direction, so it is given as: where A is a constant. (x, y) = Asin(k₁x) sin(k₂y), (a) The wavenumber k₁ in the x direction is quantized in terms of an integer n₁. Using the standing wave condition, find the possible values of k₁. (1) (b) The wavenumber k2 in the y direction is quantized in terms of a different integer n₂. Using the standing wave condition, find the possible values of k₂. (1) (c) Each state of the 2D infinite rectangular well is defined by the pair of quantum numbers (n₁, n₂). What is the energy of the state Eni,n₂? JXZ1