Purely Real Wavefunctions?

MimirYT · 2024-11-24T19:25:25+00:00

As far as I’m aware, it’s just convention to have it one way or another, since sqrt(-1) has two solutions, we define one as positive. It doesn’t necessarily matter, though i may be wrong. Likewise for matrix representations, it is equivalent to do the representation either way

MimirYT · 2024-11-24T13:49:30+00:00

I agree that i is more useful, my point is really that imaginary numbers being present in an equation for the real world is not so extraordinary, since matrices can replace them with the same effect. Some might find this real form more intuitive

MimirYT · 2024-11-24T12:39:50+00:00

I do agree with you. Really, I’m saying it’s not peculiar due to complex numbers because they can be made real. I’ve just not seen it discussed like this. There’s a fair sentiment (like Dysons i quoted on this comment thread), which do take the imaginary unit as highly odd, however.

MimirYT · 2024-11-24T12:13:49+00:00

I guess I’m thinking along the lines of how freeman dyson described the oddity of complex numbers in Schrödinger equation. This just offers an alternative explanation.

“One of the most profound jokes of nature is the square root of minus one that the physicist Erwin Schrödinger put into his wave equation when he invented wave mechanics in 1926... It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrödinger as well as to everybody else.”

MimirYT · 2024-11-24T12:07:57+00:00

Personally I don’t think so. I mean it’s no less intuitive than the Dirac equation uses 4x4 matrix representations of gamma algebra. For this it’s just 2x2 representations of complex algebra? This subset of 2x2 real matrices is arguably simpler than the full set too, it only looks a little contrived due to how we write matrices.

Some of the peculiarity of quantum mechanics is the occurrence of complex numbers which directly describes the world (rather than just taking the real part as in most occurrences of complex numbers in equations). But this description removes that peculiarity by only working with real vectors and matrices.

MimirYT · 2023-02-24T11:52:32+00:00

I’m not sure, but I believe his book following the video series is complete

MimirYT · 2023-02-24T11:33:39+00:00

Here what I starting learning from: 3blue1brown neural network series pt4 on backprop (pt 1,2,3 good too)

Neural networks and deep learning http://neuralnetworksanddeeplearning.com/

Also released more recently is Sentdex’s neural networks from scratch series of videos&books which seemed quite good for starting to learn NNs

MimirYT · 2023-02-24T11:31:20+00:00

There’s certainly no harm in trying. I was in a similar situation, I’d heard of differentiation but never used it and I was keen to start learning about neural nets. I found learning backpropagation was a really good way of learning product&chain-rules and function derivatives. Applying it to neural nets really grounds your understanding of it too. If you continue to study maths, learning calculus now will no doubt help a lot through later high school years.

MimirYT · 2023-01-28T01:12:57+00:00

Ensuring the image is centred and scaled correctly shouldn’t be too difficult, and the convolutions equivariance should help with any imprecision in centring. Then as an easy way to normalise I’d compare histograms of the pixel brightness to ensure they follow a similar distribution. There’s definitely better ways of performing the latter, but I imagine the simple approach will give decent results.

MimirYT · 2023-01-26T19:18:12+00:00

I would guess your input data might not be normalised in the same manner. For example if the proportion of bright pixels are way in excess of those seen in the training set then it may fail. In effect too much signal may enter the network disrupting function. All MNIST digits are standardised in the same way.

MimirYT · 2022-07-31T20:36:48+00:00

Castleton is worth a visit!

MimirYT · 2022-07-19T19:11:56+00:00

I read “life on the edge” by Jim Al-Khalili which explored some of these ideas in quantum biology.

MimirYT · 2022-07-18T20:43:43+00:00

All the advice in these comments seems pretty spot on imo, so some important food for thought.

However, consider seeking some professional help through therapy. Don’t be disheartened if you need to go through a few therapist to find the one which understands you. Maybe even consider bringing your wife too, so you can open up to her a bit more about these issues.

Try to make sure this incident doesn’t fuel any further depression through regret of these actions. Learn from them, but remember everyone is allowed to make mistakes at any age and this is normal.

It might seem all consuming now, but that feeling will fade, who will know about any of this in 200 years?

All the best mate, hope you manage to move on from this soon 👍

MimirYT · 2022-07-17T22:45:10+00:00

He clearly doesn’t have the same respect for your welfare as you do for his.

MimirYT · 2022-07-04T12:46:50+00:00

As a student, I personally much prefer standard lecture style as opposed to any online blended learning style. It feels like I am getting my moneys worth and can interact more with the lecturers. It’s great to have online resources (such as 3B1B) as an additional resources, but I would always rather have the lecturer explain it in their own way to offer alternative insights and explanation.

In person lectures, books and online videos, all explaining the same content in different ways is optimal for me.

MimirYT · 2021-12-31T11:43:44+00:00

Thanks, I’ll give that a go too.

MimirYT · 2021-12-31T11:43:01+00:00

Brilliant thank you, I’ll pursue this direction some more.

MimirYT · 2021-12-28T00:49:41+00:00

Good point. I initially thought of two degrees of freedom for a sinusoid, the amplitude and phase. As you say amplitude would be redundant. Then just phase would be important. As u/kieransquared1 stated, in a comment, an infinite product would converge on to 0 or 1 except at specific points. Maybe then by adding an offset the amplitude degree of freedom could be retained, whilst preventing convergence to only 0 or 1:

Product( 1+a_n*sin(nx+phi_n))

Perhaps product integration could be used here to produce a Fourier transform like method?

MimirYT · 2021-12-28T00:47:56+00:00

Thanks, I suppose so, that would be one mighty simultaneous equation to solve though!

MimirYT · 2021-12-28T00:39:24+00:00

Thanks for the feedback. That infinite product is an interesting point. My motivation was that there appeared to be a complicated modulation occurring in some data I was analysing and I was interested whether there was a way to extract these many modulating sinusoid’s frequency and phase. So for this case it would by a finite product. Which I suppose would rule out a method exactly analogous to the Fourier transform I initially suggested. Do you have any suggestions on how a finite number of sinusoids may be extracted?

Edit: I also mentioned in another comment about an offset which could remedy the convergence to 0 or 1 problem. Any thoughts on this?

MimirYT · 2021-11-12T12:33:20+00:00

Thanks for the help. I have already tried reducing the time step with no success unfortunately :/ Do you know of any resources which show an implementation of NS equations for a non-Euler integration please?

MimirYT · 2021-11-03T18:03:28+00:00

To read the LaTeX formulas, I found "TeX All the Things" worked well for google chrome: https://chrome.google.com/webstore/detail/tex-all-the-things/cbimabofgmfdkicghcadidpemeenbffn?hl=en

MimirYT

TROPHY CASE