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 v(x, t) describe a free particle and<x >- y(x, t)* x ψ(x, t) dx, show that (d3dt) < x >-0, where x is not an explicit function of t. What is the physical meaning of this...
(a) Find ψ(x, t) and P(En) at t > 0 for a particle in a one-dimensional infinite potential well with walls at x = 0 and x = a, for the following initial state. ii. ψ(x, 0) = A(exp(iπ(x − a)/a) − 1) (b) If measurement of E at 5s, finds that E = 4π^2 h(bar)^ 2 /(2ma^2 ), what is ψ(x, t) at t > 5s for the initial state?
Assuming that the trajectory corresponding to a solution x-φ(t), y = ψ(t),-oo < t < oo, of an autonomous system is closed, show that the solution is periodic. (2) Assuming that the trajectory corresponding to a solution x-φ(t), y = ψ(t),-oo
Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where k = 2π/λ and ω = 2πf. The aim of the following exercises is to show that this expression captures many of the intuitive features of waves. a) Consider a snapshot of the wave at t = 0. Use the expression to find the possible values of x at which the crests (maximum points) of the wave are located. By what distance are neighboring...
2. (3 points) Which of these operators are linear? d2 03 ψ 04p 05 ψ dr2 exp(t) (AB)V, if A and B are linear exp(A)ψ if A is linear = = oo An and the last operator can be written as exp(A) = Σ -0-.
Construct a Simulink model of the following problem 5 xdot + sin x = f (t), x(0) = 0; f(t) = -5 if ψ(t) <= -5 ψ(t) if -5<=ψ(t)<=5 5 if ψ(t)>=5 where ψ(t) = 10 sin 4t
If the ψ of air around a tree is -14 MPa, the ψ inside the stomata is -7 MPa, and the ψ of the xylem in the leaf is -5 MPa, will transpiration occur? Why or why not?
Q3) A particle in the harmonic oscillator potential has the initial normalized wave function Ψ(?, 0) = 1 /√5 [2 ?₁ (?) + ?₂ (?)] where ?1 and ?2 are the eigenfunctions of the oscillator Hamiltonian for ? = 1,2 states. a) Write down the expression for Ψ(?,?). b) Calculate the probability density ℙ(?,?) = |Ψ(?,?)| ² . Express it as a sinusoidal function of time. To simplify the result, let ? ≡ (?² ℏ)/ 2??² . c) Calculate 〈?〉...
V(x) = ㆀ other iE t 72 T where En= Given the initial state 0 Ψ(x, 0) =-sin 5 L Normalize to find A, find the (allowable) eigenvalues and their corresponding probability of obtaining therm Calculate the average energy and determine the probability of finding the system at time tin the state an
2. [16 points] What is the solution of the time-dependent Schrödinger Equation Ψ(x, t) for the solution of the time-independent Schrödinger Equation Ψ(x) = ,in (m) in the particle in the box model? Write ω =-explicitly in terms of the parameters of the problem. Explicily show that W,(Cx.t) solves the time-dependent Schrödinger Equation 2