Schrodinger equation | Derivation and how to use it
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- Опубликовано: 31 май 2024
- In this video we see how the Schrodinger equation comes out very simply from the conservation of energy.
This is the second video. Click here for the first: • The Schrodinger equati...
My twitter: Looking_glass_u
First.
Throughout these 2 videos, I kept talking about predicting the future, and that if you know the present state, you can predict the future. Does this mean quantum mechanics is deterministic? If you don’t think so, comment on where the determinism ends and the randomness starts in this theory.
Second.
Show that, for the Schrodinger equation, this is true: That it’s the same thing to go forward in time t1 plus t2 as go forward t1 and then go forward t2. Seems obvious, but explain what philosophical consequences that might have.
Third.
I said that linearity follows from the shrodinger equation. Can you see why? Try and prove it.
Fourth.
This is for the people who’ve done Quantum mechanics before and know that in this theory time evolution is unitary- yet I just went on about linearity so much in the previous video. Show that linearity plus the assumption that time evolution maps valid states to other valid states is equivalent to saying that the evolution is unitary.
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THIS WAS AWESOME!!! I lean more toward the differential equations perspective, but it's always fascinating to see it made clear from the linear algebra perspective. Thank you for this :-)
Thank you :D!!!
I am really into linear algebra, so when I can see something that way I do :P I wish DE came more naturally to me though!
Maybe we will ;)
***** Yeah... Who knows? ;-)
We'll try our best :)! Thank you for the encouragement though.
@@ScienceAsylum wow..2 youtube gaints.
I'm a dislexic carpenter, with a love for science, I often get beautifully lost on this channel! Keep up the good work! It is RUclips gold!
Thank you!! I'm really glad it heaps you foster that love for science!
It's cool that you actually go into the nitty gritty of things :) that is what makes this channel unique
Thank youu! I really like your channel btw :D
aww thank you! means so much coming from such a big RUclipsr :)
Big youtuber LOL. I'm really not. But you know, I remember how it was like starting out. That first bit (maybe half a year for me where I made heaps of videos but hardly 30 views on each) is the hardest bit. But it's such a big learning curve and that's great. I hope you enjoy it and then one day do super super well! You deserve it!
lol you are! I saw your channel on a blog post "The Best Science RUclips Channels 2016". haha if that's not big I dunno what is! I know it's taken me so long just to get 100 subscribers but you're right there is just SO MUCH to learn. But I'm really enjoying it and trying not to focus on the numbers. Thanks for the support! Hopefully I'll be as big as you one day :)
Just checked.
And you just earned a subscriber!
It's really the comment-on-a-big-channel effect.
This is an amazing video. I've listened to hours of actual physics courses and had some mathematical understanding of the equation, but after watching this it all "clicked" into an intuitive understanding for me, for the first time.
Yayyy! I felt the same way when I was writing this video too.
Great work as always Looking Glass.
Thank you :)
Commenting on a video 6 years old but... wow! Watched several videos on Schrodinger equation and this one was the best by far! I felt like I understood it.
You made this feel so intuitive and straightforward! Thank you so much!
I got so excited when I saw your channel uploaded not one, but TWO videos!
:D I really wanted to finish both as a 'sorry I've been away'.
1. The determinism ends with the collapse postulate. I believe the explanation goes like this: prior to the collapse, everything evolves (the state vector) according to a differential equation of first order in time (Schroedinger's equation), in which if you are given the initial conditions, you have the whole future of the state vector before your eyes. It is only with the introduction of the collapse postulate that chance gets into play.
2. Mathematically, (ħ=1)
T(t₁+t₂) |ψ⟩ = exp[ iH(t₁+t₂) ] |ψ⟩
= exp(iHt₁)exp(iHt₂) |ψ⟩
= T(t₁)T(t₂) |ψ⟩.
Now, a philosophical interpretation on this? I guess that the state vector evolves in a continuous and linear fashion. (I'M GUESSING!!!) :P
3. I'll enuciate what I think I have to do:
If |Φ₁⟩ and |Φ₂⟩ are solutions to the Schroedinger's equation, then the linear combination
|ψ⟩ = α|Φ₁⟩ + β|Φ₂⟩ is also a solution to the Schroedinger's equation-and the latter is said to be "linear".
Starting from the linear combination |ψ⟩,
T(t) |ψ⟩ = T(t) ( α|Φ₁⟩ + β|Φ₂⟩ )
= exp(iHt) ( α|Φ₁⟩ + β|Φ₂⟩ )
= exp(iHt) α|Φ₁⟩ + exp(iHt) β|Φ₂⟩
= α exp(iHt) |Φ₁⟩ + β exp(iHt) |Φ₂⟩
= α T(t) |Φ₁⟩ + β T(t) |Φ₂⟩.
Starting from the solutions |Φ₁⟩ and |Φ₂⟩ up to the factors α and β, respectively,
α T(t) |Φ₁⟩ + β T(t) |Φ₂⟩ = α exp(iHt) |Φ₁⟩ + β exp(iHt) |Φ₂⟩
= exp(iHt) α|Φ₁⟩ + exp(iHt) β|Φ₂⟩
= exp(iHt) ( α|Φ₁⟩ + β|Φ₂⟩ )
= T(t) ( α|Φ₁⟩ + β|Φ₂⟩ )
= T(t) |ψ⟩.
Did I just prove (If I proved anything at all) a "if and only if" statement?
Oh man, the mathematician in me is such an amateur...
4. By unitary I'm assuming T*T = 1 (star * for dagger †). So I say that, if |ψ⟩ and |Φ⟩ are both valid states (I have my questions on what exactly is a valid state) of the Hilbert space 𝓗, then
T*T ( |ψ⟩ + |Φ⟩ ) = T*T |ψ⟩ + T*T |Φ⟩
= exp(-iHt)exp(iHt) |ψ⟩ + exp(-iHt)exp(iHt) |Φ⟩
= exp(0) |ψ⟩ + exp(0) |Φ⟩
= |ψ⟩ + |Φ⟩,
which is only possible if T*T = 1, ie, T is unitary.
I'm glad this is not a quantum mechanics test, because I'm pretty sure the state vector I'm on right now has its coefficient squared approaching the unity at being wrong in everything.
1. Nice!
2. Ooh pretty!
3. The mathematician in you doesn't seem amateur at all! Nicely done.
4. There's a bit to this- but really good work!
Ahaha, I'm really impressed actually! Do you study QM?
Love your answers. Only in 3. there was no need to prove the equality starting from both sides. If you prove equality starting from one side, you have proven it starting from any side (that's the simmetry of the equality i.e. x=y is the same as y=x).
In-depth, Crisp, Visualising, Intriguing and above all Simple. Just too good. And the way she connected Classical Mechanics' Position and Velocity determinism with Quantum Mechanics, made ample clear to me why Schrodinger equation is equivalent to F = ma. I would request you to put something, say book or material, to read at the end of video. Also put some email or website to contact you about Physics.
you're the best !! I've no words to describe how amazing you are ! Really you're ineffably incredible. I wish I could give you many more thumbs up .
That was really interesting. Homework stuff:
1. I don't think the Schrödinger equation alone determines (heh) whether
physics is deterministic, since there are several possible interpretations,
some of which are deterministic and some of which are not. That said, it's hard
for me to imagine that it wouldn't be deterministic. It seems likely to me that
you should essentially be able to simulate the universe in a Turing machine,
and that doesn't work with (true) randomness. So that leaves us with things
like the Many Worlds interpretation and Pilot Wave theory, and I'm not really a
fan of non-locality... So for now, it seems most likely to me that Many Worlds
is correct, which would imply that the Universe is completely deterministic.
Although I'm not all that familiar with the different interpretations people
have come up with.
2. In lieu of a general proof, here is a proof with a wavefunction with two
possible states (although that's probably not the correct terminology):
T(t_1+t_2)|Ψ〉
= e^((iE_1(t_1+t_2))/ħ)α|A〉 + e^((iE_2(t_1+t_2))/ħ)β|B〉
= e^((iE_1t_1)/ħ+(iE_1t_2)/ħ)α|A〉 + e^((iE_2t_1)/ħ+(iE_2t_2)/ħ)β|B〉
= e^((iE_1t_1)/ħ)·e^((iE_1t_2)/ħ)α|A〉 + e^((iE_2t_1)/ħ)·e^((iE_2t_2)/ħ)β|B〉
= T(t_1)(e^((iE_1t_2)/ħ)α|A〉 + e^((iE_2t_2)/ħ)β|B〉)
= T(t_1)T(t_2)|Ψ〉
It's hard for me to imagine what it would mean for this to not be the case - it
seems to me that it would be a completely different universe, where time
doesn't progress "in a straight line" if you will.
If we assume for a moment that time is discrete, and that τ is the smallest
possible timestep, then the above not being true
would mean that any state |A〉 would first progress into state T(τ)|A〉. After
that, two states would "exist": the original state |A〉, advanced by 2τ, and
T(τ)|A〉, advanced by τ. So we would have T(2τ)|A〉 and T(τ)T(τ)|A〉. After
that, we would have T(3τ)|A〉, T(2τ)T(τ)|A〉, and T(τ)T(τ)T(τ)|A〉. So the
amount of different states would rise linearly with the number of timesteps, I
think. (Not exponential, though, because of multiplicative commutativity.)
It's even harder to imagine what this would mean with time being continuous, but I
tihnk it would imply that you couldn't simulate physics with a time-discrete
approximation, as we tend to do.
I guess you could sort of draw an analogy to the Many Worlds interpretation?
You would presumably only ever see yourself in one of those states, and it
would appear to be random in which of them you end up. I think.
3. Not sure if this correct, but.. here we go (it seems almost too easy to be
correct?)
(I will use c_n instead of e^((iE_nt)/ħ) for simplicity)
For linearity of a function, we have to proof two properties (stolen from
Wikipedia:)
• Homogeneity of degree 1: f(ax) = af(x)
• Additivity: f(x + y) = f(x) + f(y)
First, Homogeneity:
f(ax)
= T(t)(a|Ψ〉)
= T(t)(aα|A〉 + aβ|B〉)
= c_1aα|A〉 + c_2aβ|B〉
= a(c_1α|A〉 + c_2β|B〉)
= aT(t)|Ψ〉
= a(fx)
Next, Additivity:
f(x + y)
= T(t)(|Ψ_1〉 + |Ψ_2〉)
= T(t)((α|A〉 + β|B〉) + (γ|A〉 + δ|B〉))
= T(t)((α + γ)|A〉 + (β + δ)|B〉)
= c_1(α + γ)|A〉 + c_2(β + δ)|B〉
= c_1α|A〉 + c_1γ|A〉 + c_2β|B〉 + c_2δ|B〉
= (c_1α|A〉 + c_2β|B〉) + (c_1γ|A〉 + c_2δ|B〉)
= T(t)(α|A〉 + β|B〉) + T(t)(γ|A〉 + δ|B〉)
= T(t)|Ψ_1〉 + T(t)|Ψ_2〉
= f(x) + f(y)
4. Not gonna do this one, because I don't think I understand enough to know exactly
what I'm supposed to do. (I know that unitarity means that the sum/integral of
the squared lengths of the complex amplitudes doesn't change with time, but...
I'm not sure if that's enough. No idea what "valid states" are.)
Hey, it's great to here from you! Thanks for all this!
1. It's so interesting that you talk about simulating quantum mechanics with a turing machine. That's what my research is about! But yes, it's a good point on the interpretations.
2. You're proof generalised immediately if you just add some dot dot dots. So I think this counts as the full proof! I really really like your explanation there. Going to steal some of your ideas for the video.
3. Your proof that time evolution is linear is correct!! Very nicely done too.
4. No problem, we'll talk about it!
Sounds like your research is probably quite interesting..
Thanks for checking my proofs, and yeah, feel free to use anything I've said in a video :)
Yeah, I really like my research! (Even though I'm procrastinating right now by answering comments...)
And don't worry, I already did steal some of your explanation :D
thank you for your videos. I wish you were my teacher in school. you make things easy to follow and understand. keep them coming!
3:23
More of this! More!!!
I think I'm addicted
Yay! It's nice to hear from you :)
I'll make more as soon as I can!
Best video ive seen for a long time!!
Thank you so much for you loveIy and rich in information videos. :-) I would love to see a video where you talk about energy and why it is so abstract and hard to grasp as a concept. If you don't intend to make such a video any time soon, can you point out videos that discuss the points that make you feel uneasy about energy? Or do you habe a blog, where you have written down the issues you have with it?
A peer lead me to your video on Noether’s Theorem, and I’m so glad! I’m taking quantum mechanics at a very very hard institute and my professor does not explain these concepts as well as you are! Thank you!
No problem at all! I'm so glad it helped :)!
These vids are actually so good
Yay :D Thank you!
Well, The videos really help me a lot. Thank you!
Good to see that quantum mechanics uses a lot of the ideas that I'm learning in linear algebra. Now I see why the change of basis formulas are important (e.g. going from position to momentum basis). But in the future would you please explain why it is necessary to use eigenspaces and eigenvalues? As you already mentioned, the angle of the complex numbers change for any given wavefunction (or the superposition of each state), so why do we need to use eigenbasis for these problems? Thanks.
Linear algebra is the best :) Glad you can see how useful it is here.
Great question! One way to think of it is like this. Energy eigenstates are eigenstates (in the linear algebra sense) of the time evolution operator. I.e, when you apply the time evolution operator, it just puts the eigenvalue infront, but otherwise doesn't change the state. Remember that energy is the generator of time evolution. So the energy eigenstates are the eigenstates of the time evolution. This general pattern holds for any eigenbasis. Eg, momentum generates space translations, and so momentum eigenstates are the eigenstates of space translation (as we saw).
The more traditional view is to not look at these translation operators but just define the energy and momentum operators. Then the eigenstates are eigenstates of these operators. However, I think that they aren't so helpful to think of intuitively.
Quantum mechanics is deterministic in that the probability distributions evolve deterministically through time. What is non-deterministic is our particular experience of those probabilities. If you consider the brain as being part of a QM system and the electrochemical signals as the platform on which our consciousness "runs", then our experience is dictated by whatever set of entanglements our neurons find themselves in with the rest of the universe. I find this many worlds style approach much easier to swallow than wavefunction collapse.
Well explained..
Awesome! Can you explain Dr. Richard Feynman's Path-Integral Formulation.Thanks😍😍😄
One thing I've always found difficult is how to relate the different bases to one another, and how to think of the various aspects of intuitive reality (where these all feel to be only coincidentally related) to the underlying math, how to choose the right way of looking at things through a physics lens. In this example you really highlight the beginnings of this process, I would like to hear more about how you use it in a more directed/logical way, with a clearer purpose, Your excellent video on symmetries and Emmy Noether really helped to clarify some of that, if it helps to narrow down this vague request. Another example is that I understand how you can derive newtonian mechanics from the principle of least action and lagrangian, but I don't "see" the connection, for now I take it on faith.
Yes, I still find that hard to conceptualise- I think the noether's theorem stuff is the only systematic way. I would like to understand it better though, and if I do, I will make a video!
Ah! You can derive Newton's second law from lagragian mechanics. To do it, take the euler-lagrange equation and set L=mv^2/2-V. Then you'll get out the second law, with F=-dV/dt
Re newton/lagrange - I've had that derivation explained to me before, but I think it's one of those things that my brain isn't ready to accept yet, so I very quickly forgot the fagments I did get. I think it's because I don't see the connections, I don't really know how to see the world in terms of principle of least action etc (whereas for newton's equations, i have a pretty clear picture in my head)
1. In quantum physics the only random process is observation. All unobserved processes and evolution of course occurred without anybody observing it, all unobserved quantum processes are deterministic. As Leonard Susskind points out in his book, The Black Hole War, "However many times we run a process such as evolution backward and forward in time, as long as we don't interfere with it by trying to measure it, it will always turn out to be exactly the same." - Dfpolis #18 Targets in Evolution Part 1.
I have question. Le's supose a particle is in a superpositon of 3 energy-states 1,2,3 and then you measure it. So it colapses to one state for example 3. But this means the average energy goes from 2 to 3. is it now violating te conservation of energy?
sorry for bad english
I've been thinking about your question, and the answer is... I just don't know. I'd really like to figure it out and get back to you though! Thanks for asking something so thought provoking.
You said that those coefficients cant be rotated by the same amount. What about the stationary states? Thanks for your answer.
superb content
Thank you!
This is awesome
Great!!! Please make videos about Dirac equation and Lorentz summary violation.
1. does the determinism end when one is forced to collapse the wavefunction into one of its eigenstates? I mean if the SE eqn is well defined all the time (as a superposition of x y z states), won't it always be well defined until someone does a (stupid) observation to make it collapse?
1. Yes, I'd agree with that :D
This is really cool c:
doesn't the surroundings and the dimension or continuum in which an object is change the equation? dho energy is constant it's nature might change r8?
re question number 1 (homework): Non-deterministic [at quantum level]. why? Complex numbers: the sq(-1) doesn't physically manifest BUT it's square does! And hence we work with just that which is the probability. our senses can't see along the imaginary axis so to speak. this is the crux of it.
you are brilliant
Is relativity seeing space and time as not constant the reason that it's so hard to reconcile it with quantum mechanics, since quantum mechanics sees time and space as constant? Or is it just gravity? Or some other reason? And why does it have trouble with whatever it is?
Quantum mechanics works with special relativity, it leads to the Dirac equation and to relativistic quantum mechanics. General Relativity and gravity don't works with quantum mechanics.
Your vids are awesome 👀😜👀😜👀
I don't get a thing.... I love it! Great video :)
Oh no! Why?
I'm too stupid, that's why :D but I love quantum mechanics and your videos are great. Probably I will need to watch it few times and read some "bonus" material to finally comprehence it :)
@4:00 You explained the angle theta of rotation is proportional to t. But why is theta proportional to energy E too?
Just as she explained, the states can't all stay the same, the change must be different for the different energies. I guess being proportionally related is just the simplest way.
Then isn´t the fundamental building block o QM wrong? I mean, the universe is not time symmetric, apparently, so conservation of energy doesn't hold up right?
Man, I really went above my league here, for my essay for chemistry at A-level I chose to attempt to explain and use the Schrodingers equation (linking it into quantum model). I'm really struggling. Even when I have such a well presented and formatted explanation I'm can't help but feel like I'm missing a lot of information....
Great job explaining this!
This video did raise a question though, that I haven't really considered before:
How does the premise of a energy superposition not violate the law of conservation of energy? In the Many Worlds interpretation of QM (which AFAIK is equally valid as Copenhagen), wouldn't an energy measurement appear to produce more energy in one post-measurement "world" than another? Where does this energy "go"?
*Such* a good question. I need to think about that actually!
I got to sleep on it, and thought of a loophole. I think we can maintain a consistent energy through some kind of energy entanglement between two particles. In essence, the second particle takes on any "lost" energy from the first particle.
I love the superposition animation.
Thank you :D :D (I'm always so happy to hear that sort of thing because the animating takes ages)
When I was curious about General Relativity I saved up and bought the book "Gravitation" by misner, thorne, and archi-something. Is there a book about Quantum Mechanics that you have read and would recommend to someone who wants to genuinely understand all the math and theory behind it?
I like Tim's suggestion, Griffiths is a nice gentle intro. However, what I don't like about it is that it makes QM seems a bit like it's just solving the SE again and again. I like Sakurai's Modern QM quite a lot- but it's not as simple a beginner's book. I also like 'Sneaking a look at God's cards' if you want to know some of the really interesting stuff in QM- but again maybe not an intro book.
+Tim +Looking Glass Universe Thanks a lot! I'll definitely be looking into those then.
Will you explain Eigen values and Eigen states plzzz....what do they signify??
Hi, this is great, you are amazing!!
Question: can someone please explain what is the relationship between the Schrödinger wave equation in state space, and the wave like properties of particles in actual 3d space?? Or better how can we find these wave like properties in the Schrödinger equation?
Thank you! I made a video about the de broglie hypothesis that goes into this- if that helps?
If, as a wave function, a particle is in a superpositional state then the particle has different locations, speeds and initial time frames. When we measure we entangle information from the act of measurement. What does that imply? The act of measurement entangles information with the system it is measuring. What some argued as the conscious observer effecting the outcome. But understanding is not needed to entangle information. So no conscious observer needed. Plus it adds time to the overall equation... what would appear as a linear direction.
Example... when we observe a 65 million year old mountain what we actually observe is the entanglement of the reference frame of time from the measuring process itself. The mountain is in a superpositional state of being all ages and shapes and sizes it has been and will be... a wave function of the mountain. When we measure(observe) we entangle information from the state of the measurement... it's time frame. Now giving the mountain a finite time in reference to the measurement. It aged 65 million years instantly when we measure. 😉
Before the measurement the mountain was in superposition of time... it's age. The reasoning is because time changes dependent upon speed. The faster one goes the slower time flows... effectively stopping with the speed of causality... the c in E=mc². Which happens to be the speed of light. Because the mountain is in a superpositional state it is "fuzzy". Not one thing over another... except for probability(think Feynman diagrams). The inherent randomness... Schrodinger's equation. Everything is "fuzzy" until "realized"... measured. Information has to be entangled for time frames to exist in a linear direction. Which also implies that time doesn't flow in one direction and the second law of thermodynamics states. That would always end in the same result... a Boltzmann brain. And we know to question when all of our results lead back to the same one thing... cyclic thinking is seldom if ever "correct".
Thank you for this! Yes exactly! We don't need to accept collapse at all.
My understanding is that when an interaction happens, it happens with only one of the eigenfunctions with probability determined by the magnitude of the coefficients.
Hahaha "if you hate pictures", I love your videos :)
how do you solve the second question?
Hi Looking Glass, this is a fantastic video as always!
I've tried reading a little on the different interpretations of quantum mechanics, and I stumbled upon something called decoherence. What little I understood of it is that it's how quantum systems can 'leak information' into the outside world.
I was wondering if that can exlain the apparent loss of information during measurement - that is, the interaction of the particle with a measurement device causes it to interact very complexly with the outside world, in such a way that is unpredictable because we don't know the exact state of every atom in the device, but not, in fact, completely random. Therefore, it wouldn't be necessary to admit inherent randomness in our Universe.
Am I misunderstanding what quantum decoherence is? Is there an interpretation other than the Copenhagen interpretation which uses this fact as a basis for determinism?
I'm really glad you're doing reading on this stuff- it's fascinating! Decoherence is one of these things I really really want to understand better then make a video about. From what I understand you seem to be correct though!
Thanks for your answer! I'm surprised, because in general what happens is I presume something and it when I learn more about it, I find out I was actually completely wrong. I hope you never stop making videos, they're really excellent and I learn a lot thanks to them :)
Can u make a video on infinity?
I don't know answers for any questions but i understood what u said. Its great.
Vsauce have an awesome video on infinity! About how to count past infinity.
Thank you!
I will :)
and i'm subscribing
Since T(t1+t2)=T(t1)+T(t2), we can go back and forth in the time, can't we? If t1=t, and t2=-t; we can move forward t, then go back t, we're at the same point in time.
But, I'm not sure what happens if we measure it first, then move back and forth?
ohhhh, I really like where you're going with this.
The eigenvalues are complex numbers or only just real numbers?
It is easy to refute the idea that the world is made of numbers. Numbers are unchanging and unextended. The world is continually changing and extended in space and time. Nature is not made of actual numbers which exist only in minds, but of countable and measurable things. Until they are counted and measured, they are merely potential, not actual. - dfpolis, RUclips username (75 year old physicist).
Complex Numbers.
I really liked those two videos, this one in particular for the elegance of that construction. Really nice stuff.
BUT (possibly naïve question ahead)
Isn't it a bit strong to demand that the probability of measuring any energy value remain constant?
I understand the idea of not being able to tell the difference between two states by measuring the energy, but I don't see how it would violate conservation if we "only" had constant (average) energy, with different probabilities for each value that sort of cancelled out on average.
(Does that even make sense? I may be understanding a bit wonkily what "a particle's energy" is.)
Yes! This is a fantastic question. And yes, maybe it's too much. But it implies the schrodinger equation (almost!) and the scrodinger equation implies it... so nature seems to obey this strict conservation, rather than the conservation you said which I agree, sounds reasonable.
Hm, that looks like a bit of a circular argument there... In any case, "the equation implies it" seems like a weird argument: Newton's equations imply a lot of things that don't really happen, after all.
So let me ask this: has any equation/description been derived from this "weaker" conservation? And if so, what experiments does it suggest and how does it match the data?
Good point! But yes, you could very easily test this since then a state could be in energies E1+E2+... but over time evolve to something like E_av which has the same average energy. But the results from both are very very different (in the second case you'd never expect to see E1), and the results agree to great accuracy with the S.E.
I have a funny question to ask. Can Time be taken as an imaginary quantity.?
How do you derive an expression for Ĥ? Sure, kinetic plus potential energy but how do they look like?
Ĥ is the hamiltonian and is also know as the total energy of the system, therefore it's the summation of the kinetic and potential energies. Generally, the kinetic energy can be determined by the momentum squared divided by twice the mass and the potential energy varies dependent on the system, for example, the quantum harmonic oscillator potential is 0.5*omega*x^2.
Great question! The easy bit is the kinetic. I derive this is a previous video but we know what p looks like. The issue is V. It's always based on the classical V, but there are some constraints about how it can look in QM and there's a little bit of guess work there till you get one that works.
I just rewatched your video about de Broglie's equation and remembered that P^ is just the gradient (modulo constants). What bothers me is that a state instead of being a sum of eigenstates (of e.g. the position operator) it is an _integral_ of eigenstates. From what I remember from linear algebra, this is not how basis representation works. How do you justify this transition to a "continuous linear combination" ( = integral)?
Good question! An integral is basically a sum. That's the way I'd justify it :P But if you want better justifications then... you can't find them. The issue is that the position and momentum 'basis' are not rigourously bases at all. But we tend to treat them like ones because operators can be taken into the integral (same as with the sums).
Looking Glass Universe I did some research and found out that it's quite complex mathematics. It's advanced spectral theory. The same way a symmetric matrix has an orthonormal basis of eigenvectors, a self-adjoint (≈symmetric) operator on an infinte dimensional Hilbert space can be written as a multiplication operator (the multiplicator is the analogon to the eigenvalues) in the correct representation. The infinite-dimensionality causes the "representations" to be more complex than just sums. They can be infinite sums, integrals or even more complicated constructs. But the idea is the same: eigenbasis+eigenvalues.
Wow, math is interesting! :)
What I never understood is why a system can be in a supposition of different energy states. If you measure it, it has a change of being E_1 and a change of being E_2. So what is the energy of that quantum system? Isn't it predetermined because of the conservation law?
Yeah! Fantastic question! I'll have to think about it.
+Looking Glass Universe Thanks! Only had an intro to quantum mechanics so a lot still confuses me :)
I think the determinism ends at very beginning when one choose to express the state of the particle ( position, momentum energy ) as probability distribution of waves functions.
I like it!
Looking Glass Universe well... I like your videos more. :)
I must disagree. The average number of heads or tails measured from a large number of coin tosses is well-described by Gaussian distribution. However, this tells us nothing at all on whether the behaviour of each individual coin is deterministic or not. The coin appears to be well-described by basic Newton mechanics + friction, which we perceive to be deterministic. At the same time the coin is a set of atoms, which are also tossed, the coin behaviour resulting from collective average of behaviour of these atoms. The atoms in turn may or may not be deterministic, and may or may not consist of even smaller parts.
Probabilistic theory is not able to distinguish on whether it operates on truly random numbers or on a large collection of deterministic unknowns. I personally believe that true randomness is an invention of humans that either really want maths to be real, or believe that free will is impossible without it, and due to lack of better explanation, demand that it stays. There is absolutely nothing stating that in 100 years we will not find new particles, whose collective motion is responsible for all true randomness of today, much like people used to perceive chemistry before discovery of molecules and atoms
Wait but surely the energy of the particle doesn't have to stay the same? As long as the energy of something else changes so total energy is equal? Also what do you mean when you say the equation is hard to solve? Sorry if these questions are really stupid :P
Not stupid *at all*!
So yes, I should have said the particle here can't interact with something outside in a way that allows it to transfer energy!
The equation is too hard in that, no mathematical solution exists in most cases. That's true even for Newton's laws. But it means we have to approximate an answer using a computer.
Aah OK that makes sense now thank you! Also I love your videos
Yay thank you :D
In order to answer the question about whether quantum mechanics is deterministic I'd need to know how a measurement affects the future evolution of the Schrodinger equation... which is one of the questions I've had a hard time finding an answer to (at least one that I can understand).
Yes, that's true! I'll cover it (in lots of depth!) soon
Why should the angle be linear in E? You seem to only argue that it should depend on E.
0.46, are those super saiyan particles? XD
About the deterministic one.
...
Well, we could say that evolution of individual states(if that means anything), is deterministic.
But as we see, upon measurement, they may superpose in a different manner, making it random.
That's the worst I could do :)
That was my thinking as well. The collapse of the wavefunction upon measurement defines one (deterministic) state as real. Which state gets chosen is based on probability ("randomness").
I like how that's put! That is certainly a nice way to see it.
Yes! This is right... if you measure in the right basis. Say psi1 is an eigenvector of position, and psi has psi in it. psi1 evolves to T(psi1). But T(psi) need not be a eigenvector of position anymore. So if you measure position, what happens?
Because the phase of entity is local to the eye
Wow.
Your narration has gotten a lot more... sophisticated.
Plus, you sound a bit tired?
Hey, thanks for commenting! Good to hear from you.
I'm really really interested to know what you thought of the narration. I didn't try to change it on purpose too much this time- but I did do it differently. Do you like it? I wonder why I sound tired? Maybe cos I was tired? That was a rough day and it was hard to make 2 videos at once. But, maybe it wasn't that. I don't know!
Looking Glass Universe Well, it did feel good.
But I guess the Looking Glass Universe in my mind, is the one with excited (or clumsy!) narration and cute cartoons (and Physics, obviously). But, the videos are great, so yay!
"Maybe *cos* I was tired?"
Well, that's a *sin* you're working hard!
Ahaha, thank you! But working hard on this stuff is great- it's such fun to do.
Hmm, I need to think about my narration style a bit- we'll see!
Looking Glass Universe I'll be looking forward to it!
Thanks :)
It is addition
Oh good.
Now, I can atleast explain (and understand) my Facebook cover photo better
What's your fb cover photo :P?
The uglier form of Schrödinger's equation!
I was really young when I put it.
I just knew that it existed, but literally nothing more.
I think your videos make it better.
Thanks!:)
Yay :D glad you can understand it more now!
It's not deterministic considering when we measure, it collapses one of the states randomly. However, we also know the options, so it is not totally random.
That's true!
I'm desperately trying to understand what this is about becausd it's very interesting, but the fact that we have just arrived at a constantly accelerated movement in my physics class in school doesn't help. But I'll try.
DFT/PAW FTW!
Welcome back!!!
Is it that difficult 6:46? Is not /that/ difficult. It's just a linear system of PDEs, typically with no cross-terms either. For conservative systems a solution method is obvious. All non-linear equations (well, those not taught in introductory DE courses) are much more of a pain.
"Homework"!!! (i.e. fun)
At 7:47 (problem 2) are you not assuming that the two time evolution operators commute? Should it not be
T(t1+t2) = T(t2) T(t1) ?
I mean, it's quite clear, if you have T(t1) = T(Δt) multiplied with itself m times and T(t2) is T(Δt) multiplied with itself n times then T(t1+t2) = T(Δt)*T(Δt)*...(n times)*T(Δt)*...(m times) = T(t2) T(t1), where Δt is some sufficiently tiny time-step. If this is a sound way to go about then that would also mean that T(t1) and T(t2) would commute as well. But who knows, perhaps I'm off.
Q1: Time evolution QM is deterministic in that the initial wavef'n completely determines the wavef'n at some later form; that follows from uniqueness theorems of linear DEs. Likewise, everything you calculate from it is determined e.g. all the moments of the modulus squared (interpreted as a probability density fn) are determined etc. But collapse is not.
Q3: it's quite easy when writing the S.E. in the position basis; it is a sum of linear operators (second derivatives and whatever the potential happens to be)...
About question 4, I know (or rather used to know about 2 years ago but I will have to revise) the derivation that starts form a unitary time operator and then derives (a) that iħ ∂/∂t = H (b) that the Hamiltonian is Hermitian and (c) energy conservation. Is this one the one referred to?
I have a few questions that are bugging me from time to time
Does time evolution maximise the von Neumann entropy? Is it possible to calculate from iħ ∂ρ/∂t = [H,ρ] where ρ is the density operator, to conclude that is non-decreasing?
Is it possible to arrive to some description of superposition as entanglement between eigenstates, or would that be contradictory?
The reason I'm wondering is that the S.E. is very similar to a diffusion equation and, if I remember correctly, the diffusion equation gives an evolution that maximises the Shannon entropy...
Thanks! Nice to hear from you!
Even Newton's laws are hard for 3 bodies. So yeah, even though S.E isn't at all the worst case, people dedicate there live's research to approximating solutions.
2.Yes! I should write them the way you said- though of course the time evolution that I showed here does commute. I think your solution here assumes the answer implicitly- though I may be wrong. You need to show it by using the explicit form of the evolution.
1.Yes!
3. No I'm not sure what that is. I mean, a) is the implicit in the S.E, b) does follow from U being unitary, c) not sure about this one?
Time evolution (without measurement) keeps Von Neumann entropy constant. Not sure what you mean by ? Since Tr(ρ)=1 so that is =-. But what expectation are you taking?
"Is it possible to arrive to some description of superposition as entanglement between eigenstates, or would that be contradictory?" But entanglement is between objects not states, so I don't think this would make sense.
"The reason I'm wondering is that the S.E. is very similar to a diffusion equation and, if I remember correctly, the diffusion equation gives an evolution that maximises the Shannon entropy..."
Ah! I see what you mean. Good question but alas, no.
Looking Glass Universe
Whoops, I made a mistake, it should read -Tr(ρ log(ρ)) and not , the latter doesn't make sense. Also ρ is not exactly the density matrix as its rows are indexed with the basis functions instead of the pure states, as is done for the density matrix.
Perhaps that may not even be enough and instead try to posit an operator S with eigenvalues -Tr(ρ log(ρ)) but I don't know if that is consistent.
Jason93609: it actually IS that difficult numerically if the number of particles (say, electrons) is greater than about 4 or 5. This is because the many-body SE `lives' in 3N dimensions where N is the number of particles. That is: the SE for N particles is NOT a system of N equations, just one in a high-dimensional space.
luckyluckydog123
You can write it down as a system if you want. Just as you can for several non-linear DEs.
T(t1+t2) = T(t2) T(t1) implies that T(t1) and T(t2) commute:
T(t2) T(t1) = T(t1 + t2) = T(t2 + t1) = T(t1) T(t2)
I'd love to see your face. I already love your voice. It's so cute. Please don't be creeped out, it's just a compliment haha
I'm also interested in what you say, otherwise I wouldn't stick around.
What I mean is, what you do is great, you should continue! Arf I'm so awkward...
Never subtraction
Thank you for this video :)
(warning : potentially bad english in this comment)
1. Quantum mechanics is determistic until there is a measurement (where randomness is involved in the Copenhagen interpretation)
2. (Not a valid explanation) If you can predict the behavior of the particule in each state like in Newtonian physics, then T(t1+t2) = T(t1)+T(t2) like in Newtonian mechanics for each state and because states don't act on each other (if there is no measurement), therfore T(t1+t2) = T(t1)+T(t2) for the complete wave function. Don't understood what philosophical things it implies, sorry.
3 and 4 : I never did such things for "standart physics" and I absolutely don't know how to prove such statements.
Does unitary time implies that there is multiverse theory is impossible, or are all those conclusions valid only when there is no measurement involved, and in this case, what is the point of proving that time is linear/unitary if a measurement can break the proof down ?
1. Nice!
2. No, I think your answer there is philosophical! I like it.
3. Not a problem!
Yes, exactly, many worlds has to treat measurement differently. How it does it is by saying measurement doesn't collapse the wavefunction. We'll talk about that lots in future videos.
Homework:
I think this doesn't mean that QM is deterministic, in newtonian mechanics, if you knew a particle state at one time, you could also know its state at all times, in QM, the schrodinger equation tells us that if you know a particle's wavefuction at one time, you could know it at all times, but this still isnt deterministic because it doesnt change that when you actually measure that, the outcome is based on probability
looking at one eiganstate od energy E, if you go forwards in time by t, you rotate it by t multiplied by a constant, and if you go forwards in time more by T, you rotate it more, by T times the same constant, t(E/h)+T(E/h)=(t+T)(E/h) and of course that must be true, this equation describes the real world, and the in the world, waiting 3 seconds, and waiting 2s + 1s, makes no diference, this is also a consequence of linearity
Speaking of linearity, it happens because everything changes with respect to a constant, so it satisfies aditivity and homegeniety
1. I like that answer a lot!
2.Yes!
Thanks for you answers! I'm going to make a video about them soon and I'll discuss some points you've raised!
At 3,59, you say that "the only solution is to make sure that all the angles are different for each energy" (Why should they all be different ? If only one angle theta is different from the others then it becomes impossible to factorize the expression and it turns to be enough to produce a different state ). Just after, you say that "the amount of rotation should depend of how much you've gone forward in time..." and later "That suggests that the right amount of rotation is E.time".
Your derivation is based on these asumptions but they are just...asumptions. :-(
Could you be a little bit more precise on these points ? Thanks !
PS: Still in love with your magic voice ;-)
1. The angles don't *have* to be different for each energy, but in general they *can* be different.
2. Having the angles be proportional to time is pretty much forced by the requirement that T(t1)T(t2) = T(t1 + t2).
Each eigenvalue represents a different discrete level. However, we can have two or more eigenvalues being the same. These are degenerate eigenvalues and will rotate synchronously in time. Typically, the eigenvalues are different so each one rotates at a different rate wrt time. But, as stated, they cannot all be the same.
It "SEEMS" that All Areas = "ONE"....???...Does THIS ALWAYS 'ture'....??? What ABOUT Conditional Probabilities Here????
yes
I think it's hard to give an answer to determinism.
Check Veritasium's last video ruclips.net/video/WIyTZDHuarQ/видео.html
I saw it! It's great :)
Opposite of determinism is free will. to form a non-realism free quantum mechanics we need to give up determinism and take the path of free will. older physicist refuse to even look at the option, may be younger people like you will like to know about it. I have complied my ideas in a small book. will like to sent the book to you. I only need to know the postal address to which this book: Smolin Sir, I have the answers can be dispatched. S. k. Malhotra, House no 310, Sector 37A, ""
Schrodinger's equation cannot be derived from anything. It is as fundamental and axiomatic in Quantum Mechanics as Newton's Laws is in classical mechanics (we can prove the Newton's Laws as an approximation of the Schrodinger's equation in the classical level). If you scrutinize the definition given above, one will find that the relation H=T+V that is being used is nothing but the energy conservation principle. So Schrodinger's equation is actually the energy conservation principle from a quantum perspective. Just like one has no proof for the energy conservation other than experiments which always seem to satisfy it, Schrodinger's equation has no pen-and-paper proof. The only evidences of its validity are experiments that have never violated the equation till date.
Philosophical consequence of energy conservation?:- if energy is equivalent to the phase-state "temperature" of discrete-dimensional timing integration as demonstrated by physical chemistry, then it's the unitary state of time that is the fundamental constant. (By philosophical word analysis)
"Temperature" is one of the "6 impossible things" believed in before breakfast, that makes Physics easy? Chemistry is pure consequence.
Temperature is a "dumb" interpretation of actuality because it doesn't mean what the measurement of a relative state implies about real world properties. And the same approach applied to quantum duality is why the "weird" label gets pasted on the too-hard basket. (Still need the basket though)
I'm in love with you....
The randomness starts with imprecise measurement. You have to account for the evolution of every possibility that could have resulted in your measurement.
waste of time
cannot avoid counting trees and misses the forest