a) The time derivative of inner product of the two normalized solutions of Schroedinger equation is
Now using the Schroedinger equation and its complex conjugate
we get
Now the integrand is a full derivative of x, therefore it can be easily integrated to get
For a normalized wave function we must have
or else the integral
will not converge.
Hence we get
b) The normalization of the initial wave function is
From part (a) we can deduce that
this normalization is conserved. This is important because in quantum mechanics is interpreted as probability density and a probability density function should always be normalized.
c) Consider again the Schroedinger equation and its complex conjugate
Multiply (9) by and (10) by then subtract (10) from (9)
Defining
and
Equation (11) can be written as
Integrating equation (14) from a to b we get
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation,...
1. The time-dependent Schrödinger equation The time-dependent Schrödinger equation is -R2 824(1,t) + V (1,t) (1,t) = in 2m 0:2 . (a) For V1, t) = 0, show that the wave function (1,t) = A sin (kr - wt) does not satisfy the time- dependent Schrödinger equation. (b) For VI,t) = 0, Show that I, t) = A cos(kr - wt) + i sin (kr - wt) does satisfy this equation. This is a simple demonstration that the wavefunction in...
1. Prove that the time evolution of the density operator p (in the Schrödinger picture) is given by: (1) p(t) = U(t, top(tout(t, to) 2. Suppose we have a pure ensemble at t = 0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrödinger equation.
Solve the following problems HW9. Show that the time-independent Schrödinger equation is given by P(x)/(x) = Eve from the traveling wave equation and the wave function (x./)=v(x)cos or HW10. Example 9.3 Calculation of a normalization factor Given that the wavefunction for the hydrogen atom in the ground state (n = 1) is of the form = Ne , where r is the distance from the nucleus to the electron and do is the Bohr radius, calculate the normalization factor N.
(VI.1) At timet 0, the 3D wavefunction of a particle is (x, y, 2) A(x t y+z)eko (a) Determine the normalization constant A (b) What is the probability that a measurement of L2 and L will yield the results 2h2 and 0, respectively?
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
A particle in the harmonic oscillator potential, V(x) - m2t2, is at time t 0 in the state ψ(x, t-0) = A3ψο(x) +4ψι (2)] where vn (z) is the nth normalized eigenfunction (a) Find A so that b is normalized. (b) Find ψ(x,t) and |ψ(x, t)12 (c) Find x (t) and p)(t). what would they be if we replaced ψ1 with V2? (hint: no difficult calculations are required) Check that Ehrenfest's theorem (B&J 3.93) holds for this wavefunction. (d) What...
Transverse waves on a string obey the equation -where u(x,t) is the time Cu (a) Ifat t=0, the transverse displacement u(x,0) =_and__ = 0 find u(x,t)Use the 1+x Cx general solution of the wave equation. (b) Ifthe two ends ofthe string are fixed at x=0 and x=a and at t=0 u(x,0)-0, adx,t= 0) = dx(a-x) , use separation of variables to obtain u(x,t). Cx
Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and g are smooth (Co) functions. Let 7(t) be the solution of the initial value problem with γ(0) = (1,2). Prove or give a counterexample to the statement that the w-limit set of γ can contain more than one critical point. Suppose that all solutions of the differential equation f(x,y), y = g(x,y) exist for all time and that f and...
An LT-I system with the following differential equation y’(t) + 3 y(t) = x(t) has a Zero State Response of yzsr(t) = -2 exp(-5t) u(t) + 2 exp(-3t) u(t) when an input signal: x(t) = 4 exp(-5t) u(t) is applied to the system. What is the Zero State Response of the following system beginning at time t = 0 seconds, y’(t) + 3 y(t) = x’(t) -2 x(t) if the same input signal is applied to the system, and it...
Using matlab, create a 3-D plot of the wave equation y(x,t)= cos(omega*t-beta*x) .Plot for a time range from t=0 to t=2T and a space range of x=0 to x=2lambda