A particle of mass m moves freely inside a symmetric infinite well of width a. At t= 0 it has the wave function: A Va A 1) Determine A so that the wavefunction is normalized. (x,0) O 3 37x 1 5a COS Cos COS 5лх
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- The normalised wavefunction for an electron in an infinite 1D potential well of length 89 pm can be written:ψ=(-0.696 ψ2)+(0.245 i ψ9)+(g ψ4). If the state is measured, there are three possible results (i.e. it is in the n=2, 9 or 4 state). What is the probability (in %) that it is in the n=4 state?Show the relation LxL = iħL for the quantum mechanical angular momentum operator LShow that the following function Y(0,9)= sin 0 cos e eiº is the solution of Schrödinger 1 1 equation: sin 0 21 sin 0 00 Y(0,0)= EY (0,9) and find the sin 0 dp? energy, E.
- For a particle in a one-dimensional box, calculate the probability of the particle to exists between the length of 0.30L and 0.70L if n = 5.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 general solution of the Schrodinger equation for a particle confined in an infinite square-well potential (where V = 0) of width L is w(x)= C sin kx + Dcos kx V2mE k where C and D are constants, E is the energy of the particle and m is the mass of the particle. Show that the energy E of the particle inside the square-well potential is quantised.
- make a quantum scheme that performs the addition of a pair of two-qubit numbers x and y modulo 4: |x, y> → |x, x + y mod 4>A 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.Calculate the probability and probability density to find the particle between X = 0 and X = a /n when it is in the n state