Help! Physics Confuses Me: Understanding Quantum Mechanics

[Tomrocker]

Physics confuses me...

I just came to the realization that I have no clue what I'm studying.

The more I study Quantum Mechanics the more confused I get.

What's the deal with Quantum Mechanics? Right now we just started perturbation theory and it's just some little kink in a solvable potential...What the hell? Putting a bump in the bottom of a box? Why does nobody ever explain what this means?

What's the point of doing that? It's all just solving first-order, second-order corrections, nondegenerate, degenerate...degenerate, I know- two states share the same energy. A state, a wavefunction...what does that mean?

I'm still not quite sure exactly what a wave function even is...and I'm in second semester quantum mechanics!...I know it has a statistical interpretation and that makes sense, kind of.

But it comes out of some magical thing called the schrodinger equation which nobody ever bothered to explain. Where the hell did this equation come from? It's just MAGIC? WTF? I might as well believe in Zeus for christ sake.

All this dirac notation, hermitian operators, ladder operators, ...Solving for the hydrogen atom was the only thing that made any sense.

Even worse I don't think I really even know what the hell A FORCE IS! Everyone just throws around force, force, it's a force. It does this because a force is there. WTF is a force? I know there are four of them, but why are they there and why do they work the way they do?

Sigh...It's quite pitiful the amount I don't know. Does anyone know any quantum mechanics books that explains the concepts and not just how to calculate all these abstract complicated homework problems I have to do. Something to really explain why I'm solving these problems...

Should I already know this stuff by now? Maybe I missed everything along the way and got lost in the math...

-Tom

 

[simic4]

hey tom don't sweat. the patter as one learns physics is: at first feeling you know something, then realizing you dont,,then this leads to deeper understadning till you think you know something,,,then you again realize you dont,,,and each cycle leads to a deeper understanding.

perhaps you should look at a more axiomatic approach to qm.,,, to understand the assumptions...you should also understand the simple two slit expirements better... i know it sounds lame,, but this is really where one gets an awe for the wierdness of qm, and develops a sense of intuition for qm systems,,, otherwise its easy to get lost in calculations..i bet you didnt know that those egeinvectors of the position operator are not claimed to be realizable in nature,,,they are not actually in your hilbert space.. however,,any properly normalized linear combination is... same for the momentum eigenstates..

something i didnt know until a few days ago,,it came out of a feeling of being uneasy about those delta functions that come from overlapping position egeinkets..

 

[selfAdjoint]

But it comes out of some magical thing called the schrodinger equation which nobody ever bothered to explain. Where the hell did this equation come from?

1) Einstein explained the fact that the photolelectric effect is proportional to the frequency of the impinging light rather than its intensity by postulating that EM comes in quanta (later called photons), and that for a quantum of frequency f the energy is given by e = hf, where h, of course, is Planck's constant.

2) DeBroglie saw that he could explain electron statistics showing up in spectra by positing that they also satisfied e = hf and were in orbits that "waved" in the frequency f and had to come out even once around the circle, which gave quantized orbits and worked with the spectra.

3) Schroedinger said "Hey! Debroglie's idea is neat, but he only does that fixed frequency f, which means a fixed energy e. In dynamics particles interact with potentials and the enrgy will vary - be a solution of a differential equation in fact. And how would that work with e = hf?" And working through the math that is giving you the pip, he DERIVED that equation from first principles. It was the very reverse of magic, it was built up logically!

As for not knowing what the wave function "really" is, welcome to the club. Your professor doesn't know either and neither does anyone here. You won't find any lack of of people who want to sell you an interpretation, though. The wave function is what came out of Schroedinger's equation when he had it built. He originally thought it was a real field like EM, but that didn't work out. And since then the math has gone from one strength to another, but understanding hasn't kept up.

 

[masudr]

Perturbation theory

For some systems we can solve the energy-eigenvalue problem and determine all the energy eigenstates (also known as the time-independent Schrodinger equation). This includes a particle in infinite/finite potential wells, particle in quadratic potential (simple harmonic oscillator), particle in Coulombic potential (hydrogen atom) among others. However, the number of systems we can solve is severely limited.

So what we can do is take a system which we can perfectly solve (i.e. have energy eigenvalues and eigenstates), and add a small "perturbation" to it. The idea is that if the perturbation is small, then the changes in the corresponding eigenvalues and eigenstates will also be small (much like Taylor series). There are well defined procedures for obtaining the first order, second order shifts. These procedures depend on whether or not our original solvable system was degenerate or not, and also if our perturbation is time-dependent or not.

P.S. I'm sure I repeated stuff that was obvious to you, so sorry for that.

 

[Tomrocker]

I was kind of irritable/grumpy and in an uproar when I wrote the post. So I'll just say that I've calmed down a bit now.

i bet you didnt know that those egeinvectors of the position operator are not claimed to be realizable in nature,,,they are not actually in your hilbert space.. however,,any properly normalized linear combination is... same for the momentum eigenstates..

Yeah, I learned that last semester. Thank you for reminding me though, I had forgotten.

Schroedinger said "Hey! Debroglie's idea is neat, but he only does that fixed frequency f, which means a fixed energy e. In dynamics particles interact with potentials and the enrgy will vary - be a solution of a differential equation in fact. And how would that work with e = hf?" And working through the math that is giving you the pip, he DERIVED that equation from first principles. It was the very reverse of magic, it was built up logically!

Yeah, I knew it had to have been derived from something. It just seems magical to me because I do not quite understand the details of where it comes from. I suppose it would help me to read up on some of the history of quantum mechanics.

 

[simic4]

haha yeah i figured..

 

[RogerPink]

Tomrocker,

First of all, you're asking exactly the right questions. When you do physics, you should ask "what does this mean", otherwise your just doing math. Secondly, I can offer no answers, only my opinion of the questions you've asked, it's up to you to gather as many opinions as possible, read as many interpretations as possible and then develop your own educated opinion on what these things mean. Remember, Relativity came about because Einstein said to himself "what does it mean that the speed of light is constant no matter the reference frame, what does that mean?" Next thing you know, space is bending, time is stopping, twins are aging at different rates, it's chaos, chaos! Sorry. Anyway, if you like physics, never stop asking these questions.

Now, here are my opinions,

What are forces? Forces are how pieces of matter interact with each other. Without forces, matter wouldn't be matter (what would hold it together?). It would be just a bunch of fundamental particles flying through space at constant velocities floating right through each other (collisions require force). That's why force is equal to mass times a change in velocity (acceleration). It's shaking things up! That's the simple answer. There are two big fundamental answers/definitions (as far as I know), one from QFT and one from Relativity. I leave it to others to beat those horses to death.

Wavefunction - A wavefunction is related to a probability distribution (where something probably is). Heres the thing, in quantum mechanics, the uncertainty principle says that you can't know both the position and the momentum of a particle exactly at the same time, so you end up with a probability distribution of where it probably is (and how fast its probably moving). All of quantum mechanics consists of manipulating probability distributions instead of a point like in classical mechanics. That's the beauty of classical mechanics, you can say "the energy is there" and do your math. In quantum mechanics, you don't know where anything is exactly, just where it probably is. You start with where it probably is, you apply a force, you end with where it probably will be. (Igbkfte)

Schrodinger Equation- Schrodinger just took the Hamiltonian from classical mechanics (which worked with definite position and momentum) and applied it to this quantum world of probability. It even looks like the classical Hamiltonian. The classical Hamiltonian equation which was just the Kinetic Energy + the Potential Energy of a system was a clever trick that allows us to handle complicated systems by just keeping track of the energy. So things can be spining, falling, sliding, whatever, you add up the energy and you can derive equations of motion, which will tell you what the system will do. Take a good look at that Schrodinger equation, it's just the kinetic energy plus the potential energy times a wavefunction is equal to "gasp" the total energy times the wavefunction. Why times the wavefunction? Because the energy isn't all in one place, it's spread out over a probability. (Remember though, wavefunction does not equal probability, it is the math math tool we use to manipulate probabilities (It's a subtle difference)

Degeneracy- This just means that you can't tell two things apart just by looking at their energy. Here is a real world example (kind of). Imagine a ball spinning clockwise with a certain rotational speed in space, it has kinetic energy. Now imagine the same ball, in the same spot, spinning at the same speed in the opposite direction, it has the same kinetic energy as the first example, but the two balls are clearly different. So you can't tell the two apart just by looking at the energy, they are degenerate. Why does it matter? Because when we use the Hamiltonian, we are using energy to calculate. We better account for this or our answer will be off.

I'm out of time, I hope it helps. Remember, this is just my opinion, you must make your own. Nice job asking the questions though.

 

[franznietzsche]

Tomrocker,

First of all, you're asking exactly the right questions. When you do physics, you should ask "what does this mean", otherwise your just doing math. Secondly, I can offer no answers, only my opinion of the questions you've asked, it's up to you to gather as many opinions as possible, read as many interpretations as possible and then develop your own educated opinion on what these things mean. Remember, Relativity came about because Einstein said to himself "what does it mean that the speed of light is constant no matter the reference frame, what does that mean?" Next thing you know, space is bending, time is stopping, twins are aging at different rates, it's chaos, chaos! Sorry. Anyway, if you like physics, never stop asking these questions.

All of this I absolutely agree with.

Now, here are my opinions,

What are forces? Forces are how pieces of matter interact with each other. Without forces, matter wouldn't be matter (what would hold it together?). It would be just a bunch of fundamental particles flying through space at constant velocities floating right through each other (collisions require force). That's why force is equal to mass times a change in velocity (acceleration). It's shaking things up! That's the simple answer. There are two big fundamental answers/definitions (as far as I know), one from QFT and one from Relativity. I leave it to others to beat those horses to death.

I don't like the idea of dealing with forces personally.

Thinking of force as
$$\vec F = m \vec a$$

is a bad idea IMO, simply because it is not relativistically invariant. The laws of physics should be the same to all inertial observers (this is the principle of relativity). Formulating force this way breaks that principle. Much better to simply think of force as

$$\vec F = \frac{d \vec p }{d t}$$

Of course, this doesn't answer the question of what is force? Again, in IMO it is better to think of things not in terms of forces but in terms of energy.

 

[David Burke]

I don't know how much of the history you know but it really does help to understand what QM is striving toward. It basically began with the question "Whats the smallest thing that exists?". We eventually found the atom and things were looking pretty good except that classical physics couldn't describe thermal anomalies that were apparent in hot bodies. Plank then figured out that you could correct the problems by assuming energy always comes in a discrete packet with a particular value and called that the Plank constant (this is an extremely brief and generalised account). So then we dicovered that particles have a wave property through Young's double slit experiment and then what the differential equations were to describe that wave came soon after. From here the phillosophical idea of what's the smallest thing we can find sort of lost it's drive and QT took off with a drive of it's own because it became apparent that we had stumbled onto something equally important to our understanding of the universe. From the 1940's onward it's pretty much been an exploration in mathematics with only reletively recent physical delving through atom smashers and experiments in entanglement. Having said this you should read about the EPR parradox and Einsteins debates with Bhor as the description iv'e put here is very short and lacks much of the flavour of a fuller desciption. PS sorry if iv'e insulted anyone by abbreviating the history of this extremely complicated and easily miss understood topic, that was not my intention.

 

[RogerPink]

Thinking of force as
$$\vec F = m \vec a$$

is a bad idea IMO, simply because it is not relativistically invariant. The laws of physics should be the same to all inertial observers (this is the principle of relativity). Formulating force this way breaks that principle. Much better to simply think of force as

$$\vec F = \frac{d \vec p }{d t}$$

Of course, this doesn't answer the question of what is force? Again, in IMO it is better to think of things not in terms of forces but in terms of energy.

I agree, describing things in terms of energy always seems like the best way to go. Force of course being a (negative) small change in potential energy over a small distance, area, volume, etc.

 

[Daverz]

This "crisis of faith" is a natural thing. It probably occurs to many people when they get to a certain level in their studies. When I was an undergraduate I thought I knew a lot of ****. It wasn't until I got to graduate school that I realized how very, very little I really did know.

Also, I think it's expected that while you should be able to pick up the mechanics of doing Physics, a deeper understanding is a continuing project via research, seminars, teaching, etc.

An approach to QM I find fascinating is to start with measurement in experiments (e.g. Stern/Gerlach or photon polarization) and work your way up to the Dirac formalism by induction. Schwinger does this in his https://www.amazon.com/dp/3540414088/?tag=pfamazon01-20, but find I can't always follow his train of thought very well. Amazon has an excerpt (that ends about the point it starts to get really interesting, of course.) Baym does something similar in his lectures, but he's much less ambitious and more heuristic in his approach.

I'd also like to recommend, again, Chester's Primer of Quantum Mechanics, which does an excellent job motivating the Dirac formalism.

There are actually whole books on the https://www.amazon.com/dp/048640689X/?tag=pfamazon01-20. (Haven't read it.)

It also can't hurt to read over parts of your classical mechanics text that you may not have had a chance to really ponder at the time.

 

[Daverz]

I agree, describing things in terms of energy always seems like the best way to go. Force of course being a (negative) small change in potential energy over a small distance, area, volume, etc.

For conservative forces.

 

[Tomrocker]

I think I was kind of confusing the concept of force in general (F = dp/dt) with the four fundamental "forces." Or is it probably more accurate to say fundamental interactions? I guess these would be called action at a distance forces...?

As I understand there are theories for why each of these things cause a body to accelerate (or to cause a change in momentum) ...like quantum chromodynamics, QED, General Relativity...and I don't understand any of this stuff...So I don't really have an understanding of why these work or why they're there at all...

The book on Force and the QM lectures look quite interesting. I'll definitely give them a read.

 

[Gokul43201]

On force: https://www.physicsforums.com/showpost.php?p=966424&postcount=5

 

[masudr]

Interaction is a better word than force. The term force comes from Newtonian mechanics, and gets upgraded in special relativity, but (from what I know) it's better not to think in terms of forces.

Instead we have fields permeating every point in space (and time), and different fields interact with each other (and themselves) in certain ways. One of the things physicists do is try and write down how these fields interact with each other (e.g. this is what Einstein did with general relativity: he showed how the metric field and matter fields interact).

The more ambitious physicists try and answer your questions of "where they come from?" I doubt anyone has a good answer to that.

 

[jtbell]

But it comes out of some magical thing called the schrodinger equation which nobody ever bothered to explain. Where the hell did this equation come from?

Here's a description of how Schrödinger originally came up with his equation:

https://www.physicsforums.com/showthread.php?p=418069#post418069

 

[David Burke]

Yep, sounds to me like you need to get back to the ground roots. I've done a lot of background research into the hows and whys of QM and I can tell you that it does follow a logical path as it progresses to todays theories. Take the strong nuclear force. It has an amazing story behind it and simply by knowing what inspired the people involved you gain the understanding of what the purpose of the math is. I have the same problem as you, knowing the algebra is useless unless you know what path to take in your head. Unfortunately in QT you can't use the same techniques for 'understanding' classical physics when calculating solutions because none of the ideas involved are in anyway intuitive. This argument is as old as Einstien. QT is not normal (causal) and it's not supprising when you try to think of the scales involved! I mean it is a mental strain to even imagine an object as small as an atom. So what I did was tried to make it causal by understanding why we have each part of the math. What reason was behind having to have a mathematical description of the pilot wave? Why couldn't we just get along without it? The mathematical structure of QT/QM/QFT/ST etc wasn't just picked at random and you can understand the connections if you understand the problems each step overcame. Trouble is that this has taken me about four years of reading to achieve on top of learning the mechanics/math. I know you don't have that much time to loose in history leasons but in areas where your having real problems I think it might help. Funny enough but the Elegant Universe DVD's on string theory have one of the best QT outlines I've come across, simply because it's entertaining and it covers lots of the more interesting stories from QT-ST, from their you'll have a better idea which areas you might want to start from. PS theirs nothing like a good book!

 

[3trQN]

I think taking a closer look at how the theory has developed historically is allways good, its easier to keep an eye on the bigger picture that way, but sometimes it can mean you develop an affinity for other peoples ideas without fully considering all the options.

I don't know much about QM myself, but i have been in a similar position to the OP a few times.

 

[Thrice]

There's no real "why" with physics. The deeper you go the more you uncover. You can hope to link things together & maybe get some satisfaction out of that, but the real question is "how", not "why". It's all math, not religion.

 

[Chronos]

As Niels Bohr said: “If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.“

Digging through my electronic library, here are a some mostly painless selections.

http://www2.slac.stanford.edu/vvc/theory/quantum.html
http://www.math.rutgers.edu/~oldstein/papers/qts/node1.html
http://www.chemistry.ucsc.edu/teaching/switkes/CHEM163A/Fall02/KWILSON/qm/wfprobability.html

Some more detailed treatises I think are pretty good without being terribly mind numbing:

http://arxiv.org/abs/physics/0004072
http://motionmountain.dse.nl/text.html