Recent content by I_laff

Ah of course. Thanks, that cleared things up.

Apologies if the answer is obvious, but why does ## \Omega(E_1)\Omega(E_2) ## give the total number of states?

## \Omega(E_1)## is the number of microstates accessible to a system when it has an energy ##E_1## and ##\Omega(E_2)## is the number of microstates accessible to the system when it has an energy ##E_2##. I understand that each microstate has equal probability of being occupied, but could...

Thanks for the reply, my question better put is, where does the second equation, $$\vec{v}(\vec{r})=\frac{c}{x^2+y^2} \begin{pmatrix} -y \\ x \\0 \end{pmatrix}.$$ come from. Is it simply that equation because the velocity field of that equation matches the description of a free vortex (i.e...

I am aware of the equation of the velocity of a forced vortex, which is simply ## v = r\omega ##. However, the velocity for a free vortex is ## v = \frac{c}{r} ## (## c ## is the 'circulation constant', ## r ## is the radius, ## v ## is the velocity and ## \omega ## is the angular velocity). I...

For a canonical ensemble the probability of occupying a certain microstate varies depending on the energy, however I thought that every microstate has an equal chance of being occupied. So what part of the canonical ensemble have I misunderstood?

How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)

Actually would the wavefunction be ##\sqrt{\frac{2}{L}}\cos(\frac{n\pi x}{L})## instead?

If you have a particle in a 1-d box with a finite potential when ##0 < x < L ## and an infinite potential outside this region, then the normalised wavefunction used to describe said particle is ## \psi (x) = \sqrt{\frac{2}{L}}\sin(\frac{n\pi x}{L})##. However, if you had say instead a finite...

Ah, that's a nice way to think about it :ok:

I understand the properties of the Kronecker delta, what I didn't understand was why ##c_m## came out. So, let me get this right. When applying the Kronecker delta to ##\sum_{n = 1}^{\infty} c_n##, you will get a coefficient value that both ##\psi_n## and ##\psi_m## share (due to the properties...

It was from Griffiths, Introduction to quantum mechanics second edition chapter 2

I can't see how the textbook produces the following relationships between angles: $$ \theta = \phi + \alpha \qquad (1)$$ $$ 2\theta = \alpha + \alpha ' \qquad (2)$$ My thinking is that the exterior angle theorem for triangles was used to create expression ##(1)##, but I am unsure as to how...

I referred to it as a trick because that's what the material I was reading referred to it as. I'll take a look into Fourier transforms then.

I understand that the solutions to the time-independent Schrodinger equation are complete, so a linear combination of the wavefunctions can describe any function (i.e. ##f(x) = \sum_{n = 1}^{\infty}c_n\psi_n(x) = \sqrt{\frac{2}{a}} \sum_{n=1}^{\infty} c_n\sin\left(\frac{n\pi}{a}x\right)## for...