Let ψ(x, t) describe a free particle and < x> = ∫ψ(x, t)* x ψ(x, t) dx, show that (d2/dt2) < > = 0, where x is not an explicit function of t. What is the physical meaning of this second order derivative?
Let ψ(x, t) describe a free particle and < x> = ∫ψ(x, t)* x ψ(x, t) dx, show that (d2/dt2) < > = 0, where x is not an explicit function of t. What is the physical meaning of this second or...
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0, otherwise where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle a) Determine A from normalization. b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e.,...
A free particle moving in one dimension has wave function Ψ(x,t)=A[ei(kx−ωt)−ei(2kx−4ωt)] where k and ω are positive real constants. At t = π/(6ω) what are the two smallest positive values of x for which the probability function |Ψ(x,t)|2 is a maximum? Express your answer in terms of k.
3. At time t-0 a particle is represented by the wave function A-if 0 < x<a ψ(x,0) = 0 otherwise where A, a, and b are constants. a) Normalize ψ(x,0). b) Draw (x,0). c) Where is the particle most likely to be found at t-0? d) What is the probability of finding the particle to the left of a? e) What is the expectation value of x?
Condsider the ODE d2 1 (t) + 50 x (t) = F(t) dt2 where the forcing function is given by the Fourier series F(+) = 0 +21 on sin (nt) with co = 9, c1 = 10, ... Assuming a particular solution of the form Ip (t) = a0 + Anal (an cos (n t) + bn sin (nt)) find and enter the exact values of an and bn requested below. 20 41 == 61 - 10
2. Let u(z,t) be a differentiable function on R x [0, 0o). a) Show that the directional derivative of u at (x, t) = (zo, to) along v is Dvu(x, t) = ▽u(ro, to) , v b) Solve the following homogeneous linear transport equation ul + uz = 0, u(x,0) =-2 cosx c) Solve the following non-homogeneous equation ut-2uz--2 cos (x-t), u(x, 0) = sin x d) Solve the following second-order homogeneous linear euqation u(z,0) = sin x, ut (z,...
Problem 2: Time development of a free particle The wave function of a free particle at time t- 0 is given by exp(2K1T Now answer the questions below. l. what is the time evolved wave function ψ(z,t) ? points 2. What is the average momentum at any future time? 4 points 3. What is the average energy at any future time ? 3 points
Problem 2: Time development of a free particle The wave function of a free particle at...
5. A free particle has the initial wave function, where A and a are positive real constants. (a) Normalize ψ(x,0). (b) Find φ(k). (c) Construct $(z,t), in the forn of an integral. (d) Discuss the limiting cases (a very large, and a very small).
Problems binomial random variable has the moment generating function ψ(t)-E( ur,+1-P)". Show, that EIX) np and Var(X)-np(1-P) using that EXI-v(0) and Elr_ 2. Lex X be uniformly distributed over (a b). Show that EX]- and Varm-ftT using the first and second moments of this random variable where the pdf of X is () Note that the nth i of a continuous random variable is defined as E (X%二z"f(z)dz. (z-p?expl- ]dr. ơ, Hint./ udv-w-frdu and r.e-//agu-VE. 3. Show that 4 The...
(5) Do all parts (a) Let f be a function, and let >0. Write explicit forsmulas for the second difference and second central difference operators, Δ, and each of which depend on h (b) For a function u(x, t) of two variables, consider the second order partial differ- ential equation CER This is the so-called snave equation. Construct a numerical method for approx imating solutions to this equation, by using the second forward difference for the variable t, and the...
Jo (2-x) dx 12. (3pts) Let f(t) be a continuous function. Find the derivative of y = f(x).Sh(t) dt. 12