The old 'Particle in a 3D Box' type question

[Ragnarak]

We're studying QM at the minute and it's quite easy to visualise the superposition of wavefunctions in 1 dimension and not that much harder in 2 dimensions but when it comes to 3 dimensions it seems to be much harder to visualise the wave functions. The closest analogue i can think of is prolly some sort of lattice but instead of points at the intersections it's more like 'blobs'.

 

Does anyone have a better way to think of it?

[blike]

Heh, I havn't taken anything with QM yet, so it beats the heck out of me I'm not even quite sure what a superposition of a wave function is .

 

What class is this in? Is it a QM course, or some sort of physics..

 

</nonhlepful response>

[Ragnarak]

Originally posted by blike

Heh, I havn't taken anything with QM yet, so it beats the heck out of me I'm not even quite sure what a superposition of a wave function is .

 

What class is this in? Is it a QM course, or some sort of physics..

 

</nonhlepful response>

 

I'm in 2nd Year of university, tho the course is a third year course. (Without going into detail, anyone not amazingly stupid 'accelerates' and takes courses from the next year to give more choice of doing different options the year after)

 

It's a QM course within a physics degree.

[Guest Syntax]

:: This is where faf jumps in ::

[Ragnarak]

heh we believe you, honest

 

(seriously thanks for the effort at least )

[blike]

I FUCKING SPENT AN HOUR WRITING A REPLY, THEN IT SAID TOO MANY IMAGES AND MADE ALL THE PAGES EXPIRE SO I COULDN'T POST ANYTHING.

 

Be right back, laughing.

[Sayonara]

Originally posted by Ragnarak

heh we believe you, honest

 

(seriously thanks for the effort at least )

Ha ha, Schroedinger's Posting Effect in full force

[fafalone]

I banned myself last night. After blike unbanned me this morning, I removed the limit on number of images per post, so let's try this again. I gave background equations for visualization in 2D first, but I'm skipping that. This isn't nearly in depth, but still has my basic idea. Also realize that I'm a freshman who has self-taught himself everything he knows about QM

 

Begin by using the 3D Hamiltonian operator in the Shrodinger equation:

 

[-h²/2m (:pdif:²/:pdif:x² + :pdif:²/:pdif:y² + :pdif:²/:pdif:z²) + V(x,y,z)]:lcpsi:(x,y,z) = E:lcpsi:(x,y,z)

 

H(x,y,z):lcpsi:(x,y,z) = E:lcpsi:(x,y,z)

 

Now use separation of variables:

 

[H_x(x) + H_y(y) + H_z(z)][:lcpsi:_x(x) + :lcpsi:_y(y) + :lcpsi:_z(z)] = (E_x + E_y + E_z)(:lcpsi:_x(x) + :lcpsi:_y(y) + :lcpsi:_z(z))

 

From this, you should be able to see how working with the wave function in 3 dimensions is just like in 2... you're basically solving the same equation 3 times. Normalization gives:

 

:int::int::int:|:lcpsi:(x,y,z)|²dxdydz = :int:₀^a(2/a) (sin²((n_x:pi:x)/a)dx:int:₀^b(2/b) (sin²((n_y:pi:y)/b)dy:int:₀^c(2/c) (sin²((n_z:pi:z)/c)dz = 1

 

Hope this helps a little.

[Ragnarak]

Originally posted by fafalone

I banned myself last night. After blike unbanned me this morning, I removed the limit on number of images per post, so let's try this again. I gave background equations for visualization in 2D first, but I'm skipping that. This isn't nearly in depth, but still has my basic idea. Also realize that I'm a freshman who has self-taught himself everything he knows about QM

 

Begin by using the 3D Hamiltonian operator in the Shrodinger equation:

 

[-h²/2m (:pdif:²/:pdif:x² + :pdif:²/:pdif:y² + :pdif:²/:pdif:z²) + V(x,y,z)]:lcpsi:(x,y,z) = E:lcpsi:(x,y,z)

 

H(x,y,z):lcpsi:(x,y,z) = E:lcpsi:(x,y,z)

 

Now use separation of variables:

 

[H_x(x) + H_y(y) + H_z(z)][:lcpsi:_x(x) + :lcpsi:_y(y) + :lcpsi:_z(z)] = (E_x + E_y + E_z)(:lcpsi:_x(x) + :lcpsi:_y(y) + :lcpsi:_z(z))

 

From this, you should be able to see how working with the wave function in 3 dimensions is just like in 2... you're basically solving the same equation 3 times. Normalization gives:

 

:int::int::int:|:lcpsi:(x,y,z)|²dxdydz = :int:₀^a(2/a) (sin²((n_x:pi:x)/a)dx:int:₀^b(2/b) (sin²((n_y:pi:y)/b)dy:int:₀^c(2/c) (sin²((n_z:pi:z)/c)dz = 1

 

Hope this helps a little.

 

thanks for that

 

the maths is fine and i pretty much understand what i'm doing, it's just there's no easy 'mental picture' to use

[blike]

yea, cause I understand that

[fafalone]

Do they even offer QM@usf? i'd be surprised

[Radical Edward]

think of the probabilities in terms of opacity, instead of using another axis to describe them. or think in 4 spatial dimensions. your choice.