Linearity and Superposition 8. Prove that Schrödinger's equation is linear by showing that V = a,V,(x, 1) + a,V,(x, 1) is also a solution of Eq. (5.14) if Þ, and ý, are themselves solutions.
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- Consider the 1D time-independent Schrodinger equation ħ² ď² 2m dr² with the potential where to is a parameter. (a) Show that V(x) = +V(x)] = Ev v is a solution of the Schrodinger equation. ħ² mx² sech² 1 = A sech x xo (₁)For quantum harmonic insulators Using A|0) = 0, where A is the operator of the descending ladder, look for 1. Wave function in domain x: V(x) = (x|0) 2. Wave function in the momentum domain: $(p) = (p|0)You are given a free particle (no potential) Hamiltonian Ĥ dependent wave-functions = -it 2h7² m sin(2x) e = V₁(x, t) V₂(x, t) 2 sin(x)e -ithm + sin(2x)e¯ What would be results of kinetic energy measurements for these two wave-functions? Give only possible outcomes, for example, it is possible to get the following values 5, 6, and 7. No need to provide corresponding probabilities. ħ² d² 2m dx2 and two time- -it 2hr 2 m
- the ground state wavefunction of a quantum mechanical simple harmonic oscillator of mass m and frequency, which is given by: Question mw where a = the potential is V(x) = mw²x² and N is given by: N =) 9 ax² ¡Ent Yo (x, t) = Ne ze By substituting into the time-dependent Schrödinger equation, prove that the ground state energy, Eo, is given by: Eo ħw 2consider 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)(x, t) = Ae-iwt e-(mw/ħ).x² which is a solution to Schrödinger's equation. Determine the potential V(x) that is consistent with this wave function, Note: You do not have to normalize V since Schrödinger's equation is linear.