Consider a particle of energy E moving in a one-dimensional potential well V(q), such that |dv « {m(E – V)}3/2. dq Show that the allowed values of the momentum p of the particle are such that f pdg = n+ h, 2
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- ext cenb. Consider a system whose states are given in term of complete and orthonormal set of kets |1>, |2 >, [3 >,14 > as follows: 1 1p >= |1 > + 기2 > +2|3 > + 기4 > i 214> Where the kets |n > are eigenstates of an observable A defined on the system as follows: 2 A]n > = na|n > with n = 1,2,3,4 and with a a constant number. have 4) eiyen vealue 1. If A is measured, which values will be found and with which probabilities? 2. Find the expectation value of A for the state |Ø >. 3. Assume that the state 14> is found after the measurement of A. If A is measured again immediately, which states will be found and with which probabilities? 4. Find the expectation value of A if the system is in the state |4 >. 5. Assume B another observable defined on the system, which is compatible with A. Write the uncertainty inequality between A and B. 6. If B is measured, which states will be found and with which probabilities?Be *(1) the position operator for a particle subjected to a potential of a one-dimensional harmonic oscillator P mox (Ĥ =+ 2m 2 Evaluate [î(t),î(0)] Heisenberg's chart ina) Show explicitly (by calculation) that the <p> = <p>* is fulfilled for the expectation value of themomentum. b) The three expressions xp, px and (xp+px)/2 are equivalent in classical mechanics.Show that for corresponding quantum mechanical operators in the orders shown, that <Q> = <Q>* isfulfilled by one of these operators, but not by the other two.
- A neutron of mass m of energy E a,V(x) = Vo ) II. Estimate the kinetic energy of the neutron when they reach region III.Consider the one-dimensional step-potential V (x) = {0 , x < 0; V0 , x > 0}(a) Calculate the probability R that an incoming particle propagating from the x < 0 region to the right will reflect from the step.(b) Calculate the probability T that the particle will be transmitted across the step.(c) Discuss the dependence of R and T on the energy E of the particle, and show that always R+T = 1.[Hints: Use the expression J = (-i*hbar / 2m)*(ψ*(x)ψ′(x) − ψ*'(x)ψ(x)) for the particle current to define current carried by the incoming wave Ji, reflected wave Jr, and transmitted wave Jt across the step.For a simple plane wave ψ(x) = eikx, the current J = hbar*k/m = p/m = v equals the classical particle velocity v. The reflection probability is R = |Jr/Ji|, and the transmission probability is T = |Jt/Ji|. You need to write and solve the Schrodinger equation in regions x < 0 and x > 0 separately, and connect the solutions via boundary conditions at x = 0 (ψ(x) and ψ′(x) must be…Determine the expectation values of the position (x) (p) and the momentum 4 ħ (x)= cos cot,(p): 5V2mw 4 mah 5V 2 sin cot 2 ħ moon (x)= sin cot, (p)= COS at 52mo 2 4 h 4 moh (x)= 52mo sin cot.(p) COS 2 h s cot, (p) 5V2mco 2 moh 5V 2 sin of as a function of time for a harmonic oscillator with its initial state ())))