1. Consider a free, unbound particle with V (x) = 0 and the initial state wave...

1. Consider a free, unbound particle with V (x) = 0 and the initial state wave function: Ψ(x, 0) = Ae−a|x| a>0

(a) (3 pts) Normalize the initial state wave function.

(b) (5 pts) Construct Ψ(x, t).

(c) (5 pts) Discuss the limiting cases, i.e. what happens to position and momentum when a is very small and when a is very large?

Expert Solution

genius_generous answered 2 years ago

Bound state in potential. Where V(x)=inf for x<0 V(x)=0 for 0 ≤ x ≤ L V...

Bound state in potential. Where V(x)=inf for x<0 V(x)=0 for 0 ≤ x ≤ L V (x) = U for x > L Write down the Schrödingerequation and solutions (wavesolutions) on general form for the three V(x).

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0...

Consider the following wave equation for u(t, x) with boundary and initial conditions defined for 0 ≤ x ≤ 2 and t ≥ 0. ∂ 2u ∂t2 = 0.01 ∂ 2u ∂x2 (0 ≤ x ≤ 2, t ≥ 0) (1) ∂u ∂x(t, 0) = 0 and ∂u ∂x(t, 2) = 0 (2) u(0, x) = f(x) = x if 0 ≤ x ≤ 1 1 if 1 ≤ x ≤ 2. (a) Compute the coefficients a0, a1, a2, ....

Consider the “step” potential: V(x) = 0 if x ≤ 0 = V0 if...

Consider the “step” potential: V(x) = 0 if x ≤ 0 = V0 if x ≥ 0 (a) Calculate the reflection coefficient for the case E < V0 , and (b) for the case E > V0.

Consider the step potential V (x) = 0 x ≤ 0 (I) = V0 x >...

Consider the step potential V (x) = 0 x ≤ 0 (I) = V0 x > 0 (II) . (a) What does the wave function for the scattering problem with 0 < E < V0 look like in regions I and II? (Write the equation for the wave functions, including only terms with non-zero coeﬃcients.) (b) What are the boundary conditions for the wave function and its derivative at x = 0? Deﬁne constants you are using (e.g., κ, k,...

A particle is moving according to the given data v(t)=t^2 - sqrt(t), x(0) = 0, 0...

A particle is moving according to the given data v(t)=t^2 - sqrt(t), x(0) = 0, 0 ≤ t ≤ 4. • Find x(t), the position of the particle at time t. • For what values of t is the particle moving to the left? To the right? • Find the displacement of the particle. • Find the total distance covered by the particle.

The potential of a particle is given as V (x) = -Vδ (x), with V being...

The potential of a particle is given as V (x) = -Vδ (x), with V being a positive number. Find the wave function and energy of the bounded states of this particle.

A particle moves with acceleration function a(t) = 2x+3. Its initial velocity is v(0) = 2...

A particle moves with acceleration function a(t) = 2x+3. Its initial velocity is v(0) = 2 m/s and its initial displacement is s(0) = 5 m. Find its position after t seconds.

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0)...

For the wave equation, utt = c2uxx, with the following boundary and initial conditions, u(x, 0) = 0 ut(x, 0) = 0.1x(π − x) u(0,t) = u(π,t) = 0 (a) Solve the problem using the separation of variables. (b) Solve the problem using D’Alembert’s solution. Hint: I would suggest doing an odd expansion of ut(x,0) first; the final solution should be exactly like the one in (a).

Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x)...

Consider the equation uux + uy = 0 with the initial condition u(x, 0) = h(x) = ⇢ 0 for x > 0 uo for x < 0, with uo< 0. Show that there is a second weak solution with a shock along the line x = uo y / 2 The solution in both mathematical and graphical presentation before and after the shock.

The Langevin equation for a particle in 1 dimension in an external potential V (x) is:...

The Langevin equation for a particle in 1 dimension in an external potential V (x) is: mx'' =-(dV(x)/dx)-ξx'+F(t), where ξ is the friction constant, and F(t) is the random force.Take V (x) to be harmonic: V (x) = (m/2)ω^2x^2 .Find the general solution x(t) for arbitrary initial conditions and calculate the equilibrium time correlation function C(t) = <x(t)x(0)>.

Question