We have a free particle in one dimension at a time t 0, the initial wave function is V(x,0) = Ae¯r|æ| %3D where A and r are positive real constants. Calculate the expectation value (p).
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Q: We have a free particle in one dimension at a time t = 0, the initial wave function is V (x,0) = Ae…
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Q: We have a free particle in one dimension at a time t = 0, the initial wave function is V (x, 0) =…
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- Suppose we had a classical particle in a frictionless box, bouncing back and forth at constant speed. The probability density of the position of the particle in soma box of length L is given by: 0 ans-fawr (7) p(x)= 0 x L a. Sketch the probability density as a function of position b. What must A be in order for p(x) to be normalized? Remember that you are welcome to use resources to solve integrals such as Wolfram Alpha, a table of integrals etc.Calculate the uncertainties dr = V(x2) and op = V(p²) for %3D a particle confined in the region -a a, r<-a. %3DAt time t = 0 a particle is described by the one-dimensional wave function 1/4 (a,0) = (²ª) e-ikre-ar² where k and a are real positive constants. Verify that the wave function (r, 0) is normalised. Hint: you may find the following standard integral useful: Loze -2² dx = √,
- A particle with mass m is in the state mx +iat 2h V (x, t) = Ae where A and a are positive real constants. Calculate the expectation value of (p).A particle is initially prepared in the state of = [1 = 2, m = −1 >|, a) What's the expectation values if we measured (each on the initial state), ,, and Ĺ_ > b) What's the expectation values of ,, if the state was Î_ instead?A particle with mass m is in the state .2 mx +iat 2h Y(x,t) = Ae where A and a are positive real constants. Calculate the expectation values of (x).
- Consider the wavefunction Y(x) = exp(-2a|x|). a) Normalize the above wavefunction. b) Sketch the probability density of the above wavefunction. c) What is the probability of finding the particle in the range 0 < x s 1/a ?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 ())))