v (z, t) is a solution of the Schrödinger equation for a free particle of mass m in one dimension, and v (z, 0) = A exp (-z²/a²). (a) At time t = 0 find the probability amplitude in momentum space. (b) Find v(x, t).
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- Find (a) the corresponding Schrödinger equation and wave function, (b) the energy for the infinite-walled well problem of size L, (c) the expected value of x (<x>) on the interval [0,a/4], (d) The expected value of p (<p>) for the same interval and (e) the probability of finding at least one particle in the same interval. Do not forget the normalization, nor the conditions at the border.Evaluate the E expressions for both the Classical (continuous, involves integration) and the Quantum (discrete, involves summation) models for the energy density u, (v).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.
- 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)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?
- A particle with the velocity v and the probability current density J is incident from the left on a potential step of height Uo, that is, U (x) = Uo at r > 0 and U(x) = 0 at r 0.At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)