Y:0. 4 %YO NDerive an equation which isrepresent the time-dependentSchrödinger equation. *1 Add fileThis is a required questionIf Å is the position operator, and P, is the linear momentum operator,solve the following commutative, [X² ,P.] =?.1 Add fileIf the operator Â = 3 in , and the wave function Y = 8 e-4ix , findthe eigenvalue of (Â¥),Is ¥ an eigen function of the operator Â?dx1 Add filePage 4 of 5ВackNextNever submit passwords through Google Forms.العربيةالإنجليزية

Suppose a particle of mass(m) moving with velocity(v) and the potential energy (P.E) which is…

Answered: Y:0. 4 %YO N Derive an equation which…

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# quantum mechanical particde in a harmonic osci lator potential has the initial wave function y,)+4,(x), where Y. and Y, are the real wavefunctions in the ground and fist exci ted state of the harmonic osciclator Hamiltonian- for Convenience we take mzhzw= 1 for the oscillator- What ở the probabilpty den sity of finding the par ticke at x at time tza?
We will consider the Schrödinger equation in this problem as well as the analogies between the wavefunction and how boundary conditions are an essential part of developing this equation for various problems (situations). a) Write the form of the time-independent Schrödinger equation if the potential is that of a spring with spring constant k. Write the form of the time-dependent Schrödinger equation with the same potential. Briefly describe all the terms and variables in these equations. b) One solution to the time independent Schrödinger equation has the form Asin(kx). Why might it be called the wavefunction? If this form represents a wave of light, what is the energy for one photon? (Notek here stands for the wavevector and not the spring constant.) c) Why must all wavefunctions go to zero at infinite distance from the center of the coordinate system in all systems where the potential energy is always finite?
Question A3 Consider the energy eigenstates of a particle in a quantum harmonic oscillator with frequency w. a) Write down expressions for the energies of the three lowest states. b) c) Sketch the potential for this system, along with the position of the three lowest energy levels. Add to your sketch the form of the wavefunction and the probability density in the three lowest energy states. [10 marks]
1.The odd parity eigenstates of the infinte square well , with potential V = 0 in the range −L/2 ≤ ? ≤ L/2, are given by : (see figure) and have Ψn(x, t) = 0 elsewhere , for n=2 , 4 , 6 etc a) Sketch the potential of this system , including in your sketch the positions of the lowest three energy levels . Indicate in your sketch the form of the wavefunction for a particle in each of these energy levels , and state which of the wavefunctions you have drawn could be decirbed by the Ψn written above (see figure) . b) Calculate the expectation value of momentum , ⟨p⟩ for a particle with n=2 c) Calculate the expectation value of momentum squared ⟨p 2⟩ , for a particle with n = 2 . Hint : you may use the mathematical identiy sin2 x = 1/2 (1 − cos 2x) without proof .