A particle of mass m is subjected to a force F(r) = -VV(r) such that the wave function p(p, t) satisfies the momentum-space Schrödinger equation %3D (p³/2m – aV,) p(p, t) = idp(p, t)/at, %3D where h = 1, a is some real constant and V; = a* /dp? + d² /əp, + ² /Əp? . Find the force F (r).
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- Solve the time-independent Schrödinger equation and determine the energy levels and the wave function of a particle in the potential a? V (x) = Vol a + 2r2 with a = const.The wave function of a particle in a one-dimensional box of width L is u(x) = A sin (7x/L). If we know the particle must be somewhere in the box, what must be the value of A?The wave fuuction of the ground state of a harmonic oscillator of force Constant k and mass m is vo (1) = (a/m)/4 e-az'/2, a = mwy/h, w3 = k/m. %3D %3D Obtain an expression for the probability of finding the particle outside the classical region.
- QUESTION 7 Use the Schrödinger equation to calculate the energy of a 1-dimensional particle-in-a-box system in which the normalized wave function is 4' = e sin(6x). The box boundaries are at x=0 and x=r/3. The potential energy is zero when 0 < x <- and o outside of these boundaries. 18h? m h2 8m h2 36n2m none are correctA particle with mass m is moving in three-dimensions under the potential energy U(r), where r is the radial distance from the origin. The state of the particle is given by the time-independent wavefunction, Y(r) = Ce-kr. Because it is in three dimensions, it is the solution of the following time-independent Schrodinger equation dıp r2 + U(r)µ(r). dr h2 d EÞ(r) = 2mr2 dr In addition, 00 1 = | 4ar?y? (r)dr, (A(r)) = | 4r²p²(r)A(r)dr. a. Using the fact that the particle has to be somewhere in space, determine C. Express your answer in terms of k. b. Remembering that E is a constant, and the fact that p(r) must satisfy the time-independent wave equation, what is the energy E of the particle and the potential energy U(r). (As usual, E and U(r) will be determined up to a constant.) Express your answer in terms of m, k, and ħ.As6