Equations of Electron Dispersion in an E Field

[goorioles747]

Homework Statement

Given the dispersion equation of an electron in an electric field:

Homework Equations

a) calculate the velocity of the electron if k = π/a

b) If the electric field E is applied in the -x direction, derive the time
dependence of k for an electron initially at k = π/a and position x = 0.

c) Derive the time dependence of the electron velocity, v(t), and the
time dependence of the electron position, x(t).

The Attempt at a Solution

for a, do they mean the group velocity which is a function of 1/hbar * dE/dk?

 

[nickjer]

My guess is yes. It makes little sense to just calculate the phase velocity for a single frequency.

 

[goorioles747]

And would you set the E = .5mv²? Or can you not because of the uncertainty principle?

 

[nickjer]

The dispersion isn't quadratic so there is no classical kinetic energy term. I am guessing that you are giving the dispersion for an electron in a crystal. Since you are lacking in details.

 

[paranormal]

I would like to clarify that the dispersion equation of an electron in an electric field is a mathematical representation of the relationship between the electron's energy and momentum in the presence of an external electric field. It is typically written as E(k) = E0 + eEa, where E0 is the energy of the electron in the absence of the electric field, e is the electron charge, and Ea is the energy due to the electric field.

For part a, if k=π/a, then the velocity of the electron can be calculated using the group velocity equation, v = 1/hbar * dE/dk. However, it should be noted that the group velocity is a function of the electron's momentum, not the applied electric field. So, it would be helpful to know the value of k in the absence of the electric field to accurately calculate the velocity.

For part b, the time dependence of k can be derived by considering the equation of motion for the electron, which relates the change in momentum to the applied force (in this case, the electric field). This would result in an equation of the form k(t) = k0 + eEt/m, where k0 is the initial momentum of the electron at position x = 0.

For part c, the time dependence of the electron velocity, v(t), and position, x(t), can be derived using the equations of motion for velocity and position, which relate the change in these parameters to the acceleration of the electron (determined by the applied force). These equations would be v(t) = v0 + eEt/m and x(t) = x0 + v0t + (1/2)aet^2, where v0 is the initial velocity and x0 is the initial position of the electron.