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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Q: Subject: Quantum physics Please solve it. Book: Quantum mechanics by zetili 2 nd edition
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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)The eigenfunction for OHS for n=1 is of the form Vi(x) = -「网2 ep with value = "ħw mo and energy E1 = a. Write the form of the function as a solution of the Schrodinger equation for this OHS (v(x,t) b. Draw the wave function and energy levels of this OHS until n = 4. %3DFor 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 mThe sin and cos version can be derived thru Schrodinger (in terms of particle in a box) in complex exponential form. Show that C and D are constants that can be expressed in terms of A and B using Euler's formula.asap pls
- The following problem arises in quantum mechanics (see Chapter 13, Problem 7.21). Find the number of ordered triples of nonnegative integers a, b, c whose sum a+b+c is a given positive integer n. (For example, if n = 2, we could have (a, b, c) = (2, 0, 0) or (0, 2, 0) or (0, 0, 2) or (0, 1, 1) or (1, 0, 1) or (1, 1, 0).) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls in 3 boxes, and follow the method of the diagram in Example 5.Find the wave function and its energy by solving the Schrodinger equation below for the three-dimensional box.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 2
- 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)4. A particle is in the state 2 1 Y (0,0)+ V5 Y, '(0,ø) – Y (0,¢), V5 which is a superposition of the normalized eigenstates, Y;" (0,¢), of the L² operator. Calculate the value of the total angular momentum of the particle in this state. Also, calculate the expectation value of the operator L+L_ in this state.(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.