Coherent states in quantum mechanics

📝 Problems+solutions:
- Quantum harmonic oscillator I: professorm.learnworlds.com/co...
- Quantum harmonic oscillator II: professorm.learnworlds.com/co...
💻 Book a 1:1 session: docs.google.com/forms/d/e/1FA...
📚 The coherent states of the quantum harmonic oscillator are defined as the eigenstates of the lowering operator. This seemingly simple definition leads to a plethora of properties that make coherent all-important. For example, they are the states that most closely resemble the motion of a classical harmonic oscillator, and in this context they are called quasi-classical states. As another example, they play a key role in the study of quantum optics. In this video, we define coherent states and investigate some of their most basic properties.
0:00 Intro
2:41 Definition of coherent states
4:16 Coherent states in the energy basis
12:04 Time evolution of coherent states
16:19 The raising operator has no eigenstates
21:04 Wrap-up
🐦 Follow me on Twitter: / profmscience
⏮️ BACKGROUND
Quantum harmonic oscillator: • The quantum harmonic o...
Ladder operators: • Ladder and number oper...
Quantum harmonic oscillator eigenvalues: • Eigenvalues of the qua...
Eigenvalues and eigenstates: • Eigenvalues and eigens...
Representations: • Representations in qua...
Schrödinger equation: • The Schrödinger equation
⏭️ WHAT NEXT?
Quasi-classical states: • Quasi-classical states...
Displacement operator: • The displacement opera...
Coherent state wave function || Maths: • Coherent state wave fu...
Coherent state wave function || Concepts: • Coherent state wave fu...
~
Director and writer: BM
Producer and designer: MC

[@A.p.p.y]

These 22 mins >>>>>>>> my prof 3 hours lecture on coherent states :)

[@ProfessorMdoesScience]

Glad you find it useful! :)

[@GenshinoGuy]

Indeed, easy to grab intuitively!

[@sandippaul468]

This is literally one of the most beautiful videos on youtube. Another one was "But what is Fourier Transform?" Really really can't thank you enough for this Bbbrilliant and lucid explanation.🙏🙏🙏🙏🙏

[@ProfessorMdoesScience]

Thanks for your kind words! :)

[@volkerjung615]

can't stop binge watching these videos. I'm so grateful that you guys spend so much effort in making these videos. Thank you!

[@ProfessorMdoesScience]

Thanks for watching! :)

[@A.J314]

I am taking a quantum optics course with Prof. Immanuel Bloch at the Max Planck Institute of Quantum Optics, and your videos are life saving. Thanks a lot.

[@ProfessorMdoesScience]

Great to hear this! :)

[@mariak8480]

I think there’s a minor mistake at 4:40, doesn’t the the scalar product for the coefficient c_n need to be the other way around? c_n = ?

[@asmaa.ali6]

yes true. it represents the coefficient of |\alpha> for base |n> . we can do it as follows: we multiply by

[@YossiSirote]

Great video. One technical error at 4:33 c_n(a)= not , which is the conjugate.

[@suraj.k.gaikwad]

Please do videos on Concepts of Statistical Mechanics as well as Classical Mechanics. Your video clear the important concepts in Physics. Thank You so much.

[@ProfessorMdoesScience]

Thanks for the suggestion! We hope to extend the material beyond quantum mechanics, and statistical mechanics is definitely in our radar... however, it will take some time before we have the bandwidth to cover all these topics.

[@andrewdirr824]

As a university student in physics, I would like to say thank you very, very much for taking the time to make this video (and the others on your channel). I wish I had discovered your channel sooner; it is wonderful. The clarity in your lectures is the kind of clarity that I hope for my own lectures to one day have. As a student of multiple disciplines, my experience has been that physics professors can have a harder time clearly communicating. The field seems to attract more introverted types and the language in which it is "spoken" (mathematics) can, at times, require such a high degree of complexity and abstraction that it naturally becomes more difficult to verbally communicate. That is an unfortunate confluence of factors, because it is precisely the abstractness and complexity of physics that demands of its communicators that they be especially well-versed in written and spoken word, in structuring information, knowing audiences and tailoring material to them. You do a wonderful job with all of that. A true enrichment to the field. So, again: thank you very, very much.

[@ProfessorMdoesScience]

Thanks for your kind comment, glad you like it! May we ask where you study?

[@andrewdirr824]

@@ProfessorMdoesScience Outside of Cincinnati in the US.

[@satadrudas3675]

This channel is just amazing. You guys explain these topics really well.

[@ProfessorMdoesScience]

Glad you find it useful!!

[@MrOvipare]

This litterally cannot be any clearer than this. Thanks for explaining how the coherent state is an eigenstate of the annihilation operator. It also contextualized how the number states are in the energy basis, which is made up of number states that are the eigenstate of the number operator. Would it be right to think, with quantum wave-particle duality in mind, that the energy basis is the basis of "counting" in the particle-like sense and that coherent states are in a basis of "propagation" and wave-like behaviour? They describe the same "object" which is a quantum field, but in different perspectives/bias.

[@ProfessorMdoesScience]

Glad you like this! I guess your conceptualization looks appealing, although of course there are many other states in which the state propagates in time, it is just that it does also spread as opposed to the coherent states in which its shape remains fixed :)

[@noelmanuel185]

Excellent explanation..gave a better intuition abt coherent states ..thank you ma'am ❤🔥🔥🔥🔥

[@ProfessorMdoesScience]

Glad to hear that!

[@vicerto]

Thanks a lot!! Hope you guys make more videos on QM topics, i love step by step explanations like these! Greetings from Chile

[@ProfessorMdoesScience]

Glad you like it! We do plan to continue with QM for a while, so stay tuned! :)

[@felipechoy2156]

I'm in love with your Channel! It is helping me too in my last year as an undergrad physics student.

[@ProfessorMdoesScience]

Thanks for watching, and glad to hear our videos are helpful! :)

[@paulbk2322]

One of the most beautifully presented lectures on coherent States of QHO....a great learning experience

[@ProfessorMdoesScience]

Thanks for watching, and glad you like the video!

[@qbtc]

This video is so clear and at the level of a junior/senior college QM class. Well done!

[@ProfessorMdoesScience]

Glad you like it! :)

[@bhoopendragupta4782]

I am lucky to find this KZbin channel. ❤❤

[@ProfessorMdoesScience]

Glad you found us and are enjoying the videos! :)

[@asmaa.ali6]

Great! keep going on! at 21:00 the raising operator has not coherent sates as eign states because the reason you said earlier due to the form of the coherent state with sum goes to infinity but NOT because of being not hermitian .Thank you so much

[@ProfessorMdoesScience]

Yes, all I'm saying at that point is that there is no need for the raising operator to have eigenstates :)

[@themorrigan3673]

I have a question isnt the component Cn is given by < n l alpha>?? Im actually confuse since I may have read that to be the case

[@ProfessorMdoesScience]

You are absolutely correct, this is a typo in the video, cn should be , not as we write. Thanks for pointing this out!

[@sumaiyaseif3500]

What I have been looking for, thank you

[@ProfessorMdoesScience]

Glad you found us! :)

[@mtripledot8910]

Absolutely brilliant! This will helpe a ton with my undergrad project!

[@ProfessorMdoesScience]

Glad you liked it! We have a whole series on coherent states coming up, so stay tuned! :)

[@lachiehanrahan1416]

Outstanding video, this channel is exceptional.

[@ProfessorMdoesScience]

Glad you like it!

[@00pehe]

Your videos are saving me so hard! Thank you so much!

[@ProfessorMdoesScience]

Glad to hear we are helpful!

[@manujsharma1432]

Thank you for such a wonderful explanation

[@ProfessorMdoesScience]

Glad you like it!

[@mehdisi9194]

Great video. Thank you so much

[@ProfessorMdoesScience]

Thanks for watching! :)

[@anjaruls6780]

Very very helpful video

[@ProfessorMdoesScience]

Glad you think so!

[@anuman99ful]

Superb video

[@ProfessorMdoesScience]

Glad you like it!

[@hassansafari6607]

Thank you very much. Very useful.

[@ProfessorMdoesScience]

Glad you found it useful!

[@ahmedzohia4805]

This is a great video, Thanks a lot.

[@ProfessorMdoesScience]

Glad you like it!

[@Kobitaiami]

This is really great lecture about this topic 👍👍😊😊

[@ProfessorMdoesScience]

Glad you think so!

[@themorrigan3673]

I love books but they are sometimes too tedious to unravel each one thats why Im glad to watch your video on coherent states and from there somehow I get the idea presented in the book with your help!!! please continue doing this videos. Can you do a video about WKB approximation?

[@ProfessorMdoesScience]

Glad you like our approach! And we have the WKB approximation to our list (we hope to cover a full range of approximation methods).

[@themorrigan3673]

@@ProfessorMdoesScience Glad to hear it. I will definitely watch your videos alongside the references given by my professors.

[@letaowang2960]

I hardly ever write comments but your videos are really helpful, thanks a lot :)

[@ProfessorMdoesScience]

Glad you like them, and thanks for watching! :)

[@kartiksreedhar4219]

Thank you for this video. Could you also discuss the time evolution of coherent states in the Fock basis for discrete positions, like say in a Bose-Hubbard system? I suppose that would be more advanced, but it is a topic I'm personally interested in.

[@ProfessorMdoesScience]

Thanks for the suggestion, we'll add to our (increasingly long) list of possible topics :)

[@assassin_un2890]

Great video for explaining thank you

[@ProfessorMdoesScience]

Glad you liked it!

[@assassin_un2890]

@@ProfessorMdoesScience Thanks in another video you said about Hisenberg and Schrodinger picture and pointed to interaction picture but it seems interaction picture isn't made yet, IMO you can improve your collection with that going to perturbation and such that makes it complete

[@ProfessorMdoesScience]

@@assassin_un2890 Thanks for the suggestion! We are working on a large number of videos, but unfortunately our main job keeps us very busy and cannot publish as frequently as we'd like to!

[@assassin_un2890]

@@ProfessorMdoesScience Nice, Wish you Do Great on Both sides

[@thanhtrungphan4524]

Thank you so much !!!

[@ProfessorMdoesScience]

Thanks for watching! :)

[@snjy1619]

thank u so much ma'am

[@ProfessorMdoesScience]

Glad you like our approach!

[@dbf72829]

My professor didn't tell me why lowering operator's eigen state is coherent state you explained it today, thanks a bunch 😭

[@ProfessorMdoesScience]

Happy to help! :)

[@jozsefkele7858]

Really loving these videos. Learning way more than I did from Griffiths LOL.

[@ProfessorMdoesScience]

Glad you find them useful! :)

[@user-ub7bi4sz8q]

thanks!

[@ProfessorMdoesScience]

Thanks for watching!

[@AnsImran]

Bravo!

[@ProfessorMdoesScience]

Glad you like it! :)

[@johnfinn9495]

Excellent video. I have two questions. (1) Can you find coherent states for a different Hamiltonian, e.g. H=(1/2)p^2+x*4? (2) How are the coherent states for the harmonic oscillator related to the fact that for a classical Harmonic oscillator, orbits af all energies oscillate at the same frequency?

[@ProfessorMdoesScience]

(1) In general there are no coherent states for Hamiltonians other than the harmonic one. For other Hamiltonians, you can also build wavepackets, but they will in general spread over time. (2) The coherent states are made of an (infinite) superposition of energy eigenstates, you can then try to relate the associated distribution of energies to the oscillator frequency. I hope these help!

[@johnfinn9495]

If there are no coherent states for any other potential (any other Hamiltonian), how does it turn out that coherent states are so important for Quantum Optics? Is it because the equations for the electromagnetic field without interaction with charged particles is also linear?

[@juliacoggins7292]

Excellent explanation; although I had to watch it at half speed! Thanks.

[@ProfessorMdoesScience]

Glad it was helpful! We think this is the advantage of the video format: you can watch it at any speed, or indeed re-watch it as many times as necessary :)

[@adityaprasad465]

Thanks so much for your lovely videos! Here's a crazy question as I learn QFT: a coherent state rotates around the complex plane as it evolves in time, right? In a laser, stimulated emission is said to produce photons that are "in phase" with the incoming wave. But according to quantized EM, single photons (and Fock states in general) are not wavelike and thus do not *have* phases. Instead, what is happening is that they're summing to form a coherent state, which behaves like a classical EM wave. And as the coherent state forms and evolves, and rotates around the complex plane, perhaps the new photons must match _that_ phase to contribute to it? Does that sound right to you?

[@eulersfollower7140]

Please make videos on angular momentum addition

[@ProfessorMdoesScience]

We are hoping to cover angular momentum addition after we've covered spin angular momentum!

[@rayeesabdulla1916]

Awesome Video. Very clear and precise. Could you say why Coherent state is called so? Is it related to coherence in Optics?

[@ProfessorMdoesScience]

Yes, coherent states play a key role in optics, and they were initially studied by Glauber. For this reason, coherent states are sometimes also called Glauber states.

[@quantum4everyone]

But, they were first discovered by Schroedinger in 1926. He recognized the classical nature of them all the way back then, Glauber showed the connection to quantum optics.

[@barrybai5123]

Fantastic explanation! I wish my professor could teach as clearly as you do, so that I don't have to seek help on KZbin.

[@ProfessorMdoesScience]

Glad you find these useful!

[@barrybai5123]

@@ProfessorMdoesScience Thanks a lot! Subscribed!

[@lixianghe-tf4ro]

Is there an error at 4:46 ? |α>=Σ|n> so C(α) may equal to rather than ?

[@ProfessorMdoesScience]

You are absolutely correct, this is a typo and c_n(alpha) should be . Thanks for pointing this out!

[@manishamishra7989]

Can you make a video on finite and infinite potential well

[@ProfessorMdoesScience]

We do actually have a video on the infinite potential well, you can find it here: kzbin.info/www/bejne/ppO9f2uGmZ2el8U No video yet on the finite potential well, but hopefully coming at some point in the future. Hope this helps!

[@eulersfollower7140]

Have a question if you dont mind .The number operator is a hermitian operator (ie) aa*=a*a ,a* indicates a dagger .Then why is the commutator [a,a*] not zero ?

[@ProfessorMdoesScience]

Note that (a*a)* = (a)*(a*)*=a*a, which is not the same you wrote. I hope this helps!

[@eulersfollower7140]

@@ProfessorMdoesScience Yea I see that ,my bad .Thank you for your response .And thanks for these videos ,have a great day .

[@soumenkhatua1070]

For Eigen function Alpha, can we take Eigen value p like 1, 2,3.... Similarly can we modify wave function also??

[@ProfessorMdoesScience]

Thanks for watching! I am not sure I follow your first question: what do you mean by "p"? Regarding the wave function, we'll publish a video on the coherent state wave function in the next few weeks, so stay tuned! :)

[@ProfessorMdoesScience]

The videos on coherent state wave functions are now available: * A more mathematical one: kzbin.info/www/bejne/ZqGcoXtuarh_f7c * A more conceptual one: kzbin.info/www/bejne/Y37Ed5KKr6lnh5o I hope you like them!

[@workerpowernow]

thank you-this explanation makes much more sense to me than the one I was given in my graduate quantum course. However, I'm still confused about how you'd actually represent these states with concrete coefficients. For example, we write |alpha> = exp^(-|alpha|^2/2) sum_0_infinity (alpha^n/(sqrt(n!)) |n> but if we were going to calculate the corresponding coefficients cn for a given |n>, what would we use for alpha in the given expression?

[@ProfessorMdoesScience]

Glad you like it! Alpha is the quantity that tells us what the quantum state is, so in principle, you could pick any alpha you wanted to create any quantum state you wanted. A nice way to think about it is to relate it to what a classical harmonic oscillator would do. If we have a classical harmonic oscillator with amplitude x0, then choosing alpha = x0*sqrt(m*omega/2hbar) gives a coherent state whose wave function center oscillates between -x0 and x0, so it resembles the classical case. We go over this in some detail in this video: kzbin.info/www/bejne/Y37Ed5KKr6lnh5o I hope this helps!

[@ProfessorMdoesScience]

And out of curiosity, where do you go to graduate school?

[@workerpowernow]

@@ProfessorMdoesScience university of washington in seattle. Thanks for the explanation

[@richardthomas3577]

@@ProfessorMdoesScience i gather that there are no constraints on alpha -- i.e., a lowering operator can have eigenstates with any complex eigenvalue (unlike energy, which has only a countable infinity of discrete eigenvalues and eigenstates). I.e., n can only be a non-negative integer, but the set of alphas is continuous in the complex numbers. As usual, excellent video!

[@ProfessorMdoesScience]

@@richardthomas3577 Glad you like the video! And yes, alpha can in principle take any value.

[@basitakbar12345]

Can you please tell us which book do you follow for these lectures

[@ProfessorMdoesScience]

We don't follow a specific book, but books we draw inspiration from and collectively use for the videos include those by Sakurai, Cohen-Tannoudji, Shankar, and Merzbacher. I hope this helps!

[@geraldpellegrini2782]

What happens if you "measure" the energy of a coherent state? Does it "collapse" to an energy eigenstate and all motion stops?

[@user-lx4lm6qy2o]

Wonderful videos! If every university teacher could teach like you, I think human beings would have already completed commercial controllable nuclear fusion and universal quantum computer! 😆 But I have a question in 16:16, "a coherent state stays coherent at all time." What does the second "coherent" mean? As far as I know, coherent means that two waves are totally same in frequency, phase and shape. So I am quite confused. 👀 And in 11.47 "it means that coherent states of photons can be split into other independent coherent states", could you explain it more detailly or provide some references? I don't know what it means because here I only know a coherent state can collapse into energy basis with probability in Poisson distribution. Thanks!

[@user-lx4lm6qy2o]

for the first question I think I got it: it means if the state is a coherent state at start, after time evolution it still a coherent state. sorry to bother you.

[@prikarsartam]

Your contents in this channel indeed is very 'Coherent'.

[@ProfessorMdoesScience]

Good one! ;)

[@prikarsartam]

@@ProfessorMdoesScience not all channels in KZbin evolve 'Unitarily', Great Job Sir!

[@paulbk2322]

Extraordinarily beautiful

[@ProfessorMdoesScience]

Thanks for your support!

[@lukschs1]

Y nada sobre los productos internos?

[@ProfessorMdoesScience]

No explicitamente! Aunque con el resto de videos de la serie tenemos todos los ingredientes necesarios para construir el producto interno. En concreto podemos usar el producto de dos "displacement operators" de este video: kzbin.info/www/bejne/r6G1hX9_pJJ8f8U para construir el producto interno de estados coherentes.

[@narfwhals7843]

Does this imply that a system that _does_ have a maximum energy (say because the laws of physics demand that something drastic happens at a maximum energy density in a given region) can not have coherent states at all? Or is it that we can't approximate such a system as a harmonic oscillator?

[@ProfessorMdoesScience]

The discussion definitely refers to the quantum harmonic oscillator only. But more generally, not sure if there is a maximum energy? Do you have a specific example in mind?

[@narfwhals7843]

@@ProfessorMdoesScience I was thinking of gravity specifically. A black hole defines a maximum energy in any region of space, adding more energy will grow the region. I understand gravity and qm don't necessarily play well together, but I'm sure there could be other ways to limit the maximum energy.

[@ProfessorMdoesScience]

@@narfwhals7843 Thanks for the clarification. We haven't thought about this possibility, but definitely something worth pondering about.

[@chetanwaghmare3166]

what happens to the number of photons in the coherent states when it is displaced?

[@ProfessorMdoesScience]

I've answered the same question in the video on the displacement operator, but for completeness here I repeat it: A coherent state is an eigenstate of the lowering operator, so the number of photons in a coherent state is not fixed. However, you can calculate the expectation value of the number of photons in a coherent state, and for coherent state |alpha> it is |alpha|^2. You can show this easily by calculating , where n=a^dagger a is the number operator. If you have a different coherent state |alpha'>, then the expectation value of the number operator changes to |alpha'|^2. I hope this helps!

[@chetanwaghmare3166]

@@ProfessorMdoesScience Thanks for this nice explanation. But the doubt is still open. Let's say we have |alpha> state with mean photon as N. now if we displace this state by alpha further we get (|alpha+alpha> )state with mean photon as 4N right?

[@ProfessorMdoesScience]

@@chetanwaghmare3166 If I understand your notation, I think this is correct. But just in case, what you need to do to figure out the expectation value of the number of photons in state |alpha> is to calculate , where n is the number operator and is given by n=a^dagger a. If you do this in the two states you are comparing, you should get your answer. I hope this helps!

[@themorrigan3673]

I have a question is writing the coherent state in energy eigen basis considered as "construction of coherent state"?

[@ProfessorMdoesScience]

What do you mean by "construction"? In general, you can write out a quantum state in any basis, and the energy basis is typically a rather useful one, for example to study time dependence.

[@themorrigan3673]

@@ProfessorMdoesScience There is a question in one of my problem sets that says "Construct a coherent state" and thats it, I tried to clarify it with my prfessor and he hasnt gotten back with me yet.

[@ProfessorMdoesScience]

@@themorrigan3673 Do let me know if you get a clarification!

[@themorrigan3673]

@@ProfessorMdoesScience i dont know if it helps but I still cant understand what he meant he answered me saying "construct a coherent state that will be used in finding the expectation values of the momentum and position operators in that state"

[@ProfessorMdoesScience]

@@themorrigan3673 In principle one can write a coherent state in any basis. For example, the energy basis like we do in this video, or the position basis, where then it takes the form of a wave function (as discussed here: kzbin.info/www/bejne/ZqGcoXtuarh_f7c ). To calculate expectation values of position and momentum, the wave function is a useful way to do so. An alternative is to try to write the position and momentum operators in terms of raising/lowering operators, and then the best basis would be the energy basis. So, overall, I don't think there is a single answer for this question, and different strategies could work. I hope this helps!

[@abhisekpraharaj7242]

Ma'am could you please do a video on squeezed states!!!

[@ProfessorMdoesScience]

Thanks for suggestion, we'll add to our list!

[@michaelschnell5633]

At kzbin.info/www/bejne/rmHanGxuqrKsr9U you prove that the raising operator does not have eigenstates. Now I seem to remember from math lessons, that any n*n matrix in C² has n eigenvalue / eigenstate pairs (which might be duplicates), and the eigenvalues are the roots of the characteristic polynomial, that is grade n and in C does have n roots (which might be duplicates). Now this of course holds for finite n and with an operator being an "infinite Matrix" this seems to be different such as no eigenstates / eigenvalues might exist. Is there more to that (e.g. a finite non zero number of linear independent eigenstates might exist for such an operator) ? OTOH the lowering operator does have an eigenvalue 0. I seem remember that an eigenvalue 0 implies that for a finite n*n matrix in C², there are not n linear independent eigenvectors, meaning there is no eigenbase. Does this also not hold with "infinite" operators ? I assume that an operator with an eigenvalue zero still is not invertible, Are there more implications following from that ? Thanks for more enlightenment !

[@ProfessorMdoesScience]

This is a very interesting and insightful question, and a full answer would require much more than a comment. However, to get you started I would suggest looking at the first answer in this post on StackExchange: physics.stackexchange.com/questions/445144/eigenstates-of-the-creation-operator I hope this helps!

[@dantefernandez2455]

First!

[@ProfessorMdoesScience]

Nice! :)

[@tomkerruish2982]

Personally, I prefer the non-normalized eigenstates, which lead to the (Segal-)Bargmann(-Fock) space, wherein every state becomes a nice holomorphic function. Even Dirac delta functions become Gaussians, although strictly speaking they're not in the space.

[@ProfessorMdoesScience]

Thanks for your insights!

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[@jacobvandijk6525]

SHE IS PURE. HE IS NOT.