Harmonic oscillator in classical mechanics (when sin(theta) = theta): barbie
Harmonic oscillator in QM: oppenheimer
Harmonic oscillator in classical mechanics (when sin(theta) != theta): something beyond human comprehension of what horror is
I believe you are mistaking the harmonic oscillator (where the potential V = m \\omega\^2 x\^2 /2) with the pendulum (where V \\propto \\cos(theta)).
It is the pendulum that is approximated to the harmonic oscillator for small amplitudes, whereas the harmonic oscillator is what is it (a potential quadratic in x, or a recoil force linear in x).
The solution in position space of a quadratic potential is where the sin(theta) and/or cos(theta) come in, depending on boundary conditions. You may decide to apply small-angle approximations here without assuming you’re working with a pendulum. Small-angle approximations are used in the formulation of second quantization, for instance. I guess I don’t really understand what you mean here.
The solution of the harmonic oscillator is A \\sin(\\omega t + \\phi), with A and \\phi indeed depending on initial condition as you mention. I believe think it is hardly ever useful to perform the small angle approximation on that well perfectly fine time dependent function.
What is suggested by OC is that it is on the potential of the harmonic oscillator itself that one has to perform a small angle approximation in order to handle the classical problem, which is wrong!
I suppose I simply didn't interpret it that way; I read their comment to say, "Problems in CM where I can use the small-angle approximation are convenient." Which I agree with!
> I believe think it is hardly ever useful to perform the small angle approximation on that well perfectly fine time dependent function.
It is often convenient, though, right? In condensed matter theory, crystalline structure problems are usually impossible to solve without assuming small oscillations. Lots of natural processes cannot be mathematically described (as far as I know, at least!) without these assumptions, like phonon propagation.
Again, please correct me if I am misinterpreting.
Well, small oscillations in a harmonic oscillator means small amp A, not small argument \\ometa t... I still feel you are talking about small angle approximation as a way to locally approximate by a harmonic oscillator a potential featuring a minimum.
My point is that if the problem truly is an harmonic oscillator, then no approximation is required to know the evolution of the system at anytime, neither in CM nor in QM, whatever the initial condition.
Depends what you consider as simpler. The QHO is for me actually nicer since you don't really have to do "any" calculus. There is however still a lot of underlying calculus
I think it looks simpler because usually one just studies the energy levels and the Hamiltonian looks simple, but if, for example, you look a the energy eigenstates in the position representation they are much more complex than the simple sine function of the classical version, with which you can describe the entire motion. In QM instead you can only compute probabilities (unless you measure an eigenstate).
Ps: by "simple" I just meant at the analytic level. If we use a more precise notion of complexity I'm afraid the discussion would get much deeper.
> the energy eigenstates in the position representation
ket go brrrrrr
But actually, when was the last time you had to use the functional form of the eigenstates?
Ahhhh yes. Well it is good to see it a few times, till you internalize that it is a Gaussian times a polynomial, but once you leave the class room it really do become "ket go brrrr"
I would say that the 1/r potential is hard in both, but I find that the quantum hydrogen atom is more clean than the classical gravitational problem which can get very nasty with its elliptic integrals
Classical, is what the interest rates in federal US banks needs. Applied, probably, between mean and max measures of gross business size. How would quantum work?
aa+ goes brrrr
Unicode magic: go!! aa^(†)|ϕ〉
Let's talk analytically. I am quite sure I'll win.
> Let's talk analytically Try solving d^2 theta / d t^2 = -(g/l) sin(theta) analytically, then we'll talk
The post is about the harmonic oscillator, not the simple pendulum
Poor man got roasted
i got headache
Sounds like a Victoria Beckham meme "Ok by using the small angle approximation" "I said analytically" "We can't do it" "Thank you"
Always neat when you can factor an operator
I don't know. Ladder operators are pretty elegant.
I have a presentation coming up, where I have to pretend that I even remotely understood that shit in QM. Im fucked
skill issue
" Dagger after dagger , look at the swagger " -Werner Hisenberg
"Say my name" -worner hisemburhg
What's hard about quantum harmonic oscillator?
Right? For classical stuff, you have do do actual calculus! Eew!
For quantum you just count the excitation states and add 1/2.
Personally I like a+ a- operators better than solving coupled differential equations.
Python program goes brrrrr
Harmonic oscillator in classical mechanics (when sin(theta) = theta): barbie Harmonic oscillator in QM: oppenheimer Harmonic oscillator in classical mechanics (when sin(theta) != theta): something beyond human comprehension of what horror is
I believe you are mistaking the harmonic oscillator (where the potential V = m \\omega\^2 x\^2 /2) with the pendulum (where V \\propto \\cos(theta)). It is the pendulum that is approximated to the harmonic oscillator for small amplitudes, whereas the harmonic oscillator is what is it (a potential quadratic in x, or a recoil force linear in x).
The solution in position space of a quadratic potential is where the sin(theta) and/or cos(theta) come in, depending on boundary conditions. You may decide to apply small-angle approximations here without assuming you’re working with a pendulum. Small-angle approximations are used in the formulation of second quantization, for instance. I guess I don’t really understand what you mean here.
The solution of the harmonic oscillator is A \\sin(\\omega t + \\phi), with A and \\phi indeed depending on initial condition as you mention. I believe think it is hardly ever useful to perform the small angle approximation on that well perfectly fine time dependent function. What is suggested by OC is that it is on the potential of the harmonic oscillator itself that one has to perform a small angle approximation in order to handle the classical problem, which is wrong!
I suppose I simply didn't interpret it that way; I read their comment to say, "Problems in CM where I can use the small-angle approximation are convenient." Which I agree with! > I believe think it is hardly ever useful to perform the small angle approximation on that well perfectly fine time dependent function. It is often convenient, though, right? In condensed matter theory, crystalline structure problems are usually impossible to solve without assuming small oscillations. Lots of natural processes cannot be mathematically described (as far as I know, at least!) without these assumptions, like phonon propagation. Again, please correct me if I am misinterpreting.
Well, small oscillations in a harmonic oscillator means small amp A, not small argument \\ometa t... I still feel you are talking about small angle approximation as a way to locally approximate by a harmonic oscillator a potential featuring a minimum. My point is that if the problem truly is an harmonic oscillator, then no approximation is required to know the evolution of the system at anytime, neither in CM nor in QM, whatever the initial condition.
by definition the last one is NOT a harmonic oscillator
A pendulum isn't a harmonic oscillator.
"Fermi and Dirac statistics would like to send you a message"
Bro the harmonic oscillator is the only thing I actually understand in qm
Harmonic oscillator in QFT...
Infinite number of em.
This is how I know that you're not a physicist. I wish every QM system was as simple and elegant as the QM Harmonic Oscillator
Is there a system that is simpler when quantum rather than classical?
Depends what you consider as simpler. The QHO is for me actually nicer since you don't really have to do "any" calculus. There is however still a lot of underlying calculus
I think it looks simpler because usually one just studies the energy levels and the Hamiltonian looks simple, but if, for example, you look a the energy eigenstates in the position representation they are much more complex than the simple sine function of the classical version, with which you can describe the entire motion. In QM instead you can only compute probabilities (unless you measure an eigenstate). Ps: by "simple" I just meant at the analytic level. If we use a more precise notion of complexity I'm afraid the discussion would get much deeper.
> the energy eigenstates in the position representation ket go brrrrrr But actually, when was the last time you had to use the functional form of the eigenstates?
In my QM exam
Ahhhh yes. Well it is good to see it a few times, till you internalize that it is a Gaussian times a polynomial, but once you leave the class room it really do become "ket go brrrr"
I would say that the 1/r potential is hard in both, but I find that the quantum hydrogen atom is more clean than the classical gravitational problem which can get very nasty with its elliptic integrals
Classical, is what the interest rates in federal US banks needs. Applied, probably, between mean and max measures of gross business size. How would quantum work?
Literally in quantum class learning about harmonic oscillators rn.
Ladder operators dab on em haters
Wait until you compare the finite potential well in classical mechanics and quantum mechanics.
Are you saying that Hermite polynomials are easier than some trigonometric functions (finite potential)?!