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TFM

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## Homework Statement

A Bose-Einstein condensate can be described by a wave function

[tex] \psi(x,t) = \sqrt{\rho(x,t)}e^{i\phi(x,t)} [/tex]

Where the functions:

[tex] \phi(x,t) [/tex] and [tex] \rho(x,t) [/tex]

are real.

a)

What is the probability density

b)

Calculate the probability current density as a function of ρ(x,t)and ϕ(x,t) and their derivatives.

c)

Use the results of (a) and (b) and write down the continuity equation in terms of ρ(x,t)and ϕ(x,t) and their derivatives.

d)

Substitute ψ(x,t)=√(ρ(x,t) ) e^iϕ(x,t) into the Schrödinger equation for a one-dimensional particle moving in a potential V (x). (Watch out when taking the derivatives. You need to apply product and chain rules.) Then multiply the whole equation by ψ^* (x,t) in order to simplify it. The resulting equation can be split into two: one for the real and one for the imaginary part. Take the imaginary part and compare the equation you obtain to the result you got under (c).

## Homework Equations

Probability Density:

ρ= ψ* ψ

Current Density:

[tex] j(x,t)=\frac{\hbar}{2im}\left(\phi^* \frac{d\phi}{dx}-\phi\frac{d\phi^*}{dx}\right) [/tex]

Continuity Equation:

[tex] \nabla \cdot J = -\frac{\partial}{\partial t}\rho(x,t) [/tex]

## The Attempt at a Solution

I have a lot of workings out, So I have attached them as a word document, if that is okay.

I have found (a) to be:

[tex] \rho = \rho(x,t) [/tex]

Which is the right answer.

For (b), I have:

[tex] j(x,t) = \frac{\hbar \rho(x,t)\phi'(x,t)}{m} + \frac{\hbar \rho'(x,t)}{2im} [/tex]

Now I have inserted these into the continuity equation, but I have got:

[tex] \nabla \cdot J = \frac{\hbar}{m}\rho(x,t)\phi''(x,t) + \frac{\hbar}{m}\rho'(x,t)\phi'(x,t) + \frac{\hbar}{2im} \rho''(x,t) [/tex]

Whereas for the other side:

[tex] -\frac{\partial}{\partial t} \rho{x,t} = -\rho'{x,t} [/tex]

Pleas not that the working out for b is on the first word document, and for the c and d are on the second word document.

Any ideas where I may have gone wrong?

Thanks in advanced,

TFM

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