A particle of mass m moves inside a potential energy landscape U (2) = X|2| along the z axis. Part (a) What are the units of the constant X? Part (b) If the particle has kinetic energy me at the origin at z = 0, where are the classical turning points of the motion?
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- 2) Consider a particle in a three-dimensional harmonic oscillator potential V (r, y, z) = 5mw²(r² + y² + z®). The stationary states of such a system are given by ntm(r, y, z) = vn(x)¢r(y)v'm(2) (where the functions on the right are the single-particle harmonic oscillator stationary states) with energies Entm = hw(n +l+m+ ). Calculate the lifetime of the state 201.Derive the following energy formula for a distinguishable particle system.The difference of the scalar potential squared and the modulus of the vector potential squared, Φ2 - |A|2, is Lorentz invariant (a Lorentz scalar). why the statement true?
- A potential function is shown in the following with incident particles coming from -0 with a total energy E>V2. The constants k are defined as k₁ = 2mE h? h? k₂ = √√2m (E - V₁) h² k3 = √√2m (E - V₂) Assume a special case for which k₂a = 2nπ, n = 1, 2, 3,.... Derive the expression, in terms of the constants, k₁, k2, and k3, for the transmission coefficient. The transmis- sion coefficient is defined as the ratio of the flux of particles in region III to the inci- dent flux in region I. Incident particles E>V₂ I V₁ II V2 III x = 0 x = aShow that the spherical harmonics Y2,2(θ,φ)= ((15/32π)^1/2)*sin(2θ)*e^∓2iφ and Y3,3(θ,φ)= ((35/64π)^1/2)*sin(3θ)*e^∓3iφ are normalized.Consider a particle of mass m moving in the following potential V(x)= v1 for xv2. What is the grown state energy? And the normalized ground sate funcion
- Q.3 What is zero-point energy? If a classical oscillator has energy 1/2 ℏ w, what is its amplitude?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…c) How does the classical kinetic energy of the free electron compare in magnitude with the result you obtained in the previous part?
- Consider a potential barrier represented as follows: U(x) = 0 if x < 0; εx if 0 < x < a; 0 if x > a Determine the transmission coefficient as a function of particle energy.Consider a particle of mass m moving in 1-dimension under a piecewise-constant po- tential. In region I, that corresponds to x 0. In region II, that corresponds to x > 0 the potential energy is V1(x) = 0. The particle is shot from = -∞ in the positive direction with energy E > Vo > 0. See the figure in the next page for a representation of V(x) as a function of x. Also shown in the graph (green dashed line) the energy E of the particle. (a) Which of the following functions corresponds to the wavefunction 1(x) in region I? (a1) Aeikiæ + Be-iki¤ ; (а2) Ае\1 + Bе-кӕ (a3) Aeikræ (а4) Ве- кта (b) Which of the following functions corresponds to the wavefunction 1(x) in region II? (b1) Сеkп* + De-ikr (62) C'e*I1* + De-*1¤(d) Prove that for a classical particle moving from left to right in a box with constant speed v, the average position = (1/T) ff x(t) dt = L/2, where T L/v is the time taken to move from left to right. And = : (1/T) S²x² (t) dt L²/3. Hint: Only consider a particle moving from left x = 0 to right x = L = and do not include the bouncing motion from right to left. The results for left to right are independent of the sense of motion and therefore the same results apply to all the bounces, so that we can prove it for just one sense of motion. Thus, the classical result is obtained from the Quantum solution when n >> 1. That is, for large energies compared to the minimum energy of the wave-particle system. This is usually referred to as the Classical Limit for Large Quantum Numbers.