Help with Quantum Mechanics and Continuity Equation

AI Thread Summary
The discussion focuses on solving a homework problem involving a Bose-Einstein condensate described by a wave function. Key points include the calculation of probability density and current density, leading to the formulation of the continuity equation. Participants identify algebraic errors in the current density calculations and clarify the relationship between the continuity equation and the wave function. The conversation also emphasizes the importance of applying product and chain rules when substituting the wave function into the Schrödinger equation. Overall, the thread highlights the complexities of quantum mechanics calculations and the collaborative effort to resolve misunderstandings.
  • #51
So now I need to compare this:

i\left(\frac{\hbar}{2}\frac{\partial \rho}{\partial t}\right) = i\left(\frac{\hbar^2\rho \phi''}{2m} + \frac{\phi'\rho''\hbar^2}{4m} + \frac{\hbar^2 \phi''\rho''}{4m} \right)

to:

-\frac{\partial}{\partial t}\rho(x,t) = \frac{\hbar}{m}\rho(x,t)\phi''(x,t) + \frac{\hbar}{m}\rho'(x,t)\phi'(x,t)

Well is we

\frac{\partial \rho}{\partial t} = \frac{\hbar\rho \phi''}{m} + \frac{\phi'\rho''\hbar}{2m} + \frac{\hbar \phi''\rho''}{2m}

They don't seem to similar...?
 
Physics news on Phys.org
  • #52
Oh. Your previous expression was actually incorrect. The last 2 terms in the rhs should have contained single derivatives of both rho and phi, and not double derivativs (compare with the expression prior to that).
 
  • #53
\frac{\partial \rho}{\partial t} = \frac{\hbar\rho \phi''}{m} + \frac{\phi'\rho'\hbar}{2m} + \frac{\hbar \phi'\rho'}{2m}

so,

\frac{\partial \rho}{\partial t} = \frac{\hbar\rho \phi''}{m} + \frac{\phi'\rho'\hbar}{m}

Now this does look more similar...
 
Back
Top