3. The normalized solution of the time-dependent Schrödinger equation in the one-dimensional syst...
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.
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
(a) Write down the Schrödinger equation in Dirac notation. (2%) (b) Write x-representation in one dimensional systems. (3%) down both time dependent and time independent Schrödinger equations in
1 Time-independent Schrödinger equation (TISE) Remember the (one-dimensional) time-independent Schrödinger equation (TISE) for a state ( definite energy E: with Now shift the potential energy by a constant: V(x) -> V(x) Vo Show that (a) The allowed energies (El,Ea. . .) are all shifted by Vo (b) The corresponding states (vi (x),P2( r),...) remain the same.
(a) At time t 0, a one-dimensional bound system is in a state described by the normalized wave function V(r,0). The system has a set of orthonormal energy eigenfunctions (), 2(x),.. with corresponding eigenvalues E, E2, .... Write down the overlap rule for the probability of getting the energy E when the energy is measured at time t 0 (b) Suppose that a system is described by a normalized wave function of the form (,0) an(r), where the an are...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
A particle of mass m is bound by the spherically-symmetric three-dimensional harmonic- oscillator potential energy , and ф are the usual spherical coordinates. (a) In the form given above, why is it clear that the potential energy function V) is (b) For this problem, it will be more convenient to express this spherically-symmetric where r , spherically symmetric? A brief answer is sufficient. potential energy in Cartesian coordinates x, y, and z as physically the same potential energy as the...
*Please, answer all the literals and be detailed with the answer (do all the procedure and calculations) *Do it with a clear letter Homework (scattering) 1. Consider the time dependent Schrödinger equation written in the form where 0 2mo As it is well known the temporal evolution of a wave function ψ( t) known at a specific time t is uniquely determined for all future times t, > t as well as for all past times t' < t. Moreover,...
A normalized wave function would not be of much use if it did not stay normalized. Indeed, it can be shown that time evolution of the wave function via the Schrödinger equation does not change the norm of the wave function: -00 So that the total probability density is a constant of the motion. However, there are cases where this fact is computationally inconvenient, and one would like to introduce a mechanism for the wave function to decay (or leave...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...