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I. Show that ψ-Aei(kx-ut) is a solution to the time dependent 1-D Schrodinger Equation for a free...
17. Does the solution to the time-dependent Schrodinger equation for a free particle satisfy the classical nondispersive wave equation? Use mathematics to justify your answer.
2. [16 points] What is the solution of the time-dependent Schrödinger Equation Ψ(x, t) for the solution of the time-independent Schrödinger Equation Ψ(x) = ,in (m) in the particle in the box model? Write ω =-explicitly in terms of the parameters of the problem. Explicily show that W,(Cx.t) solves the time-dependent Schrödinger Equation 2
4. Differential equation. Show that if ψ(x) is a solution of the one-dimensional time-independent Schrödinger equation, then c ψ(x), where c is an arbitrary complex constant, is also a solution.
b. The 1-D time-independent Schrodinger equation for the stationary states of a harmonic oscillator is given by: Assuming that the simplest solution is the Gaussian wave function of the form: 4(x) = Copa Show that: a mo and E no
6. Plane Wave Suppose jb' = eilkr-wt). (Note that o= eik-i-wt) satisfies the free particle Schrodinger equation.) (a) If y satisfies the Schrodinger equation with given potentials A and V, find the potentials for ' that satisfy the Schrodinger equation for a charged particle in the presence of an electromagnetic field. (b) Interpret the potential obtained.
9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish by calculating the following quantities. (Use the following as necessary: A, K, x, ,t, h, and m.) momentum Need Help?Read ItTalk to a Tutor 9. 1.66 points Show that the wave function ψ-A ei(kx-at) is a solution to the Schrödinger equation, given below, where k-2π / λ and U-0. 2m dz2 Accomplish...
1. [3 marks] Show that in general, under stationary conditions where V(r) is time inde- pendent, the solutions Ψ(,t) of the tine dependent Schrodinger equation can be written as Ψ(r, t-/(r) . g(t), what does g(t) look like for a constant total energy E = T+ V
Prove the following function (using -i) is a solution to the Schrodinger equation and determine its energy. 1/2 8Tt 1/2 8Tt
2. Schrodinger equation In quantum mechanics, physical quantities cor- respond to Hermitian operators. In particular, the total energy of the system corresponds to the Hamiltonian operator H, which is a hermitian operator The 'state of the system' is a time dependent vector in an inner product space, l(t)). The state of the system obeys the Schrodinger equation We assume that there are no time-varying external forces on the system, so that the Hamiltonian operator H is not itself time-dependent a)...
1. Give the Schrodinger equation for free electrons. Explain all terms: Hamiltonian, potential. 2. Solve the Schrodinger equation and find the wavefunction and the energy. 3. Draw the energy dispersion.