Eigenstates of the quantum harmonic oscillator

[@rimon9697]

dang......from trying to make sense of bra-ket notation to being able to understand all of your videos in a single take, i have come a long way and this is all thanks to you two great teachers. This is the first channel i have found that explains qm in a mathematically rigourous way but doesn't leave you scratching your head with more questions that you started with. Keep up the good work. Awaiting for your new videos.

[@ProfessorMdoesScience]

Really nice to hear this, really motivates us to keep going. And well done on your learning of QM! :)

[@vfazio13]

Second this! I'm currently studying for my intro to quantum mechanics course and this channel is my saving grace

[@senkum80]

what a great way to prove how particle is acting on different energy level.

[@ProfessorMdoesScience]

Thanks for watching, and glad you like it! :)

[@factos8766]

I am really enjoying these videos. mathematics used in QM is expained so easily. Thanks for these awesome creation as my concepts are growing after these videos.

[@ProfessorMdoesScience]

Glad you like the videos :)

[@IsomerSoma]

Concise and useful.

[@ProfessorMdoesScience]

Great you find it useful!

[@enricolucarelli816]

Wow! Each and every video of Professor M acts like a raising operator in my brain, with the difference that the quanta are really big! 😁 Thank you so much! 🙏

[@ProfessorMdoesScience]

Good one! :)

[@sambhavishukla2770]

when maths is explained in this way .. it just gets so easy

[@ProfessorMdoesScience]

Thanks for the kind words, and glad you like it! :)

[@rithin4538]

Thankyou so much for the videos 🙏 They helped a lot

[@ProfessorMdoesScience]

Glad they were helpful!!

[@MrFilipebento]

Very good! Congratulation. But the "Parity" link at "Background" in description is not working correctly.

[@ProfessorMdoesScience]

Thanks for this, it should be corrected now!

[@justanAverageTash]

did i just understand wat the zettili book trying to teach me in Chapter 4 in just 4 videos , how did i just find this channel

[@ProfessorMdoesScience]

Glad you've found us! :)

[@canyadigit6274]

I have a question. A particle in a closed system is in a superposition state of all possible stationary state wave functions. Upon measurement, the superimposed wave function collapses to a state with energy E_0. If it is measured again it returns the energy E_1. Does this mean that energy has been created? Doesn’t it violate conservation of energy? Does conservation of energy just no longer hold at these very small scales and only the probability of getting a particular energy is conserved?

[@ProfessorMdoesScience]

Good question! A full answer would require a few videos (some are in the pipeline), but here are a few thoughts. First, let's consider the evolution of the system without performing measurements. The state is a superposition state, and therefore it does not have a well-defined value of the energy, in the sense that you can get any value when you perform a measurement. However, there is still a conservation law for a closed system: the expectation value of the Hamiltonian is conserved while you don't do a measurement. The conservation of the expectation value is actually the definition of a "constant of motion" in quantum mechanics (which could be the energy or some other observable). Next, let's consider what happens when you perform a measurement. The superposition state collapses to some energy eigenstate, so the state of the system changes and the energy takes a specific value. It may appear that energy is not conserved, but when you perform a measurement you no longer have a closed system, and overall energy should still be conserved.

[@canyadigit6274]

@@ProfessorMdoesScience so when you measure it you’re giving it that energy?

[@ProfessorMdoesScience]

I am not sure I would say that you are "giving it" that energy. Perhaps a better way of saying it is that the superposition state is compatible with a range of energies, and which energy it has is "chosen" by the act of measuring. I hope this helps!

[@sayanjitb]

At around 5:29, you said that 1/sqrt(n !) factor was there to ensure that the | n> state is normalized as it is obtained from | 0 > is itself normalized. But instead, this factor was obtained by recursive substitution | n-i > states into the |n> state expression. Then why did you say that the factor 1/sqrt(n !) was there for normalization purpose? TIA

[@ProfessorMdoesScience]

Good point! This statement builds from a result that we derived in the video on ladder operators (kzbin.info/www/bejne/gZOcpap9mZdohpI). In that video, we show that, if |n> is normalized, then for |n+1> to also be normalized we need to build it as: |n+1>=[1/sqrt(n+1)]a^dagger |n> We then simply use this relation to build the state |n> from the state |0>, and every time we go from |i> to |i+1> we need to add a factor of 1/sqrt(i+1) to ensure that the state is normalized. I hope this helps!

[@sayanjitb]

@@ProfessorMdoesScience Thanks again.

[@armalify]

Amazing. Are you going to address the subject of Quantum Many-Body Physics in the near future?

[@ProfessorMdoesScience]

We definitely want to build from our current videos on the fundamentals of quantum mechanics and explore new topics. We have made a start at many-body quantum mechanics with our series on second quantization: kzbin.info/aero/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb But we are hoping to expand on this to look at, for example, how these ideas apply to condensed matter systems. Unfortunately, it will probably take some time to get there because we first want to finish with the fundamentals of quantum mechanics...

[@hershyfishman2929]

Great channel! Can you please explain (or link to a discussion on) the notation ⟨x | 0⟩ you used at 7:42, where the state (as a function of x) is represented as an inner product (with x as a bra vector)?

[@ProfessorMdoesScience]

Glad you like it! What we mean by this notation is that we are working with the ground state (represented by the ket |0>) and then write it down in the "position representation" by projecting it onto the position basis, using the bra

[@hershyfishman2929]

Thank you!!

[@sandeepchauhan6836]

❤ great vidoe

[@ProfessorMdoesScience]

Thanks!

[@AhsanAli-gu4bm]

thanks mam

[@ProfessorMdoesScience]

Thanks for watching!

[@thejamesmom]

I still don’t get it

[@ProfessorMdoesScience]

What specifically don't you get? The video builds on some background knowledge that you can find in our other videos, perhaps that would help?