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)>.

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genius_generous answered 2 years ago

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.

An infinite potential well in one dimension for 0 ≤ x ≤ a contains a particle...

An infinite potential well in one dimension for 0 ≤ x ≤ a contains a particle with the wave function ψ = Cx(a − x), where C is the normalization constant. What is the probability wn for the particle to be in the nth eigenstate of the innite potential well? Find approximate numerical values for w1, w2 and w3.

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal...

If V (dimension k-1) is a subspace of W (dimension K), and V has an orthonormal basis {v1,v2.....vk-1}. Work out a orthonormal basis of W in terms of that of V and the orthogonal complement of V in W. Provide detailed reasoning.

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y)...

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y) = ( 0, 0 ≤ x ≤ L & 0 ≤ y ≤ H ∞, otherwise. (a) Write down the time-independent Schr¨odinger equation for this case, and motivate its form. (2) (b) Let k 2 = 2mE/~ 2 and rewrite this equation in a simpler form. (2) (c) Use the method of separation of variables and assume that ψ(x, y) = X(x)Y (y). Rewrite...

A particle of mass ? moves in one dimension along the ?- axis. Its potential energy...

A particle of mass ? moves in one dimension along the ?- axis. Its potential energy is given by ?(?) = ??3 − ??, where ? and ? are positive constants. (a) Calculate the force on the particle, ?(?). Find the position of all equilibrium points and identify them as stable or unstable. (b) Draw an energy diagram showing the potential energy U, the kinetic energy K, and the total mechanical energy E for bound motion. Show the location of...

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).

1.- A) A particle of total energy 9V0 is incident from the -x-axis on a potential...

1.- A) A particle of total energy 9V0 is incident from the -x-axis on a potential given by V(x) = 8V0 for x < 0, V(x) = 0 for 0 < x < a and V(x) = 5V0 for x > a. Find the probability that the particle will be transmitted on through to the positive side of the x-axis, x > a. B) Consider a particle incident from the left on a potential step which is defined as V(x)...

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?

The velocity function of a particle moving along a line is given by the equation v(t)...

The velocity function of a particle moving along a line is given by the equation v(t) = t2 - 2t -3. The particle has initial position s(0) = 4. a. Find the displacement function b. Find the displacement traveled between t = 2 and t = 4 c. Find when the particle is moving forwards and when it moves backwards d. Find the total distance traveled between t = 2 and t = 4 e. Find the acceleration function, and...

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.

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