/sci/ - Science & Math

>>8624189
http://vocaroo.com/i/s0n9QSJXVICJ

This gives the manifestation of the quantum mechanical commutation relations:
Let [math] p_ { \Delta t} (t) = m \frac { x ( t+ { \Delta t})-x(t)} { { \Delta t}} [/math]. If the limit
[math] \lim_ { \Delta t \to 0}p_ { \Delta t}(t) [/math] exists, then for auxiliary
[math] \delta[/math], we have [math] \lim_ { \Delta t \to}x(t+ \delta^2 { \Delta t})=x(t) [/math].

Hence, for ever smaller time grid size [math] \Delta t[/math], e.g. an expression like
[math] x(t+ \delta_1^2 { \Delta t}) \,x(t+ \delta_2^2 { \Delta t}) \,x(t+ \delta_3^2 { \Delta t}) [/math]
converges to [math] x(t)^3 [/math].

However, for [math] x(t+ \Delta t) \approx x(t)+ \kappa { \sqrt { \Delta t}}[/math]. We find
[math] x(t+ \delta^2 { \Delta t}) \,p_ { \Delta t}(t)= \delta^4 \,m \, \kappa^2+x(t) \,p_ { \Delta t}(t)[/math].

The result says that two naively equivalent approximation schemes (e.g. [math] \delta=0 [/math] vs. [math] \delta=1 [/math]) systematically differ by an additive diffusion term (e.g. [math]m \kappa^2 [/math] here).
I.e. the Itō integral is defined with most left of the grid cells as in the explicit Euler-method numerical approximation scheme. The implicit Euler-method simply corresponds to a different notion of integral here and would give a different result.

In quantum mechanics, the difference of the product above is [math] m \, \kappa^2=m \frac {i \hbar} {2m}= \frac {i \hbar} {2} [/math].

Finally, coming to fractional quantum mechanics… we had

[math] \frac { d} { d t} \psi = \kappa^2 \frac { d^2} { d x^2} \psi [/math]

(note the imbalance of dimensions, t vs. [math]x^2 [/math]) and in turn

[math] P( \Delta x) \propto \exp \left(c \frac {( \Delta x)^2} { \Delta t} \right) [/math]

as next-step distribution, and then

[math] \langle |x| \rangle \propto t^ {1/2} [/math]

gives the non-smooth curve.

>>7767597
tl;dr Quantum leaps happen randomly but your lack of knowledge increases with time
(the time being computed back to when you did your last measurement T0).
If you measure more frequently (which always resets T0 to a more recent time), you are prone to prevent a leap.

You have a system in one of two possible states, called A resp. B. You don't know which it is. If you measure it, you know which, say A.
Then, not caring for that system anymore, according to the dynamics of the theory, the system evolves again. The probability for it to transition to the other state B grows with time
(because the probabilities in quantum mechanics change continuously, that's what the Schrödinger equation is).
If you wait a long time, it might be that next time you measure it it's in state B. However, if you measure shortly after the first time, it's probably still in state A. If you measure all the time, it most likely stays in A as long as you do your autistic repeated measurement.

Analogy I just made up:
Say you have a machine with a spring with a heavy ass weight bouncing up and down (weight bouncing = state A). After a year experience shows the spring breaks (weight doesn't do shit = state B). After 3 years you can be pretty much certain it already broke. Some of those springs, in the past, have even broken after 8 months.
Your wageslave task to look for the machine. The springs are actually cheap and so whenever you check, you replace the old spring with a new one, just in case. Now if you check the spring every 9 months, you might mostly be good (the weight keeps on swinging, state A), but it might break (transition to state B) and you lose your job. If instead you check the spring every month you can be pretty sure the system stays in state A.

>>7767597
You have a system in one of two possible states. A, B. You don't know which. If you measure it, you know which, say A. Then, not caring for that system anymore, according to the dynamics of the theory, the system evolves again. The probability for it to transition to the other state B grows with time (because the probabilities in quantum mechanics change continuously, that's what the Schrödinger equation is). If you wait a long time, it might be that next time you measure it it's in state B. However, if you measure shortly after the first time, it's probably still in state A. If you measure all the time, it most likely stays in A as long as you do your autistic repeated measurement.

Analogy I just made up:
Say you have a machine with a spring with a heavy ass weight bouncing up and down (weight bouncing = state A). After a year experience shows the spring breaks (weight doesn't do shit = state B). After 3 years you can be pretty much certain it already broke. Some of those springs, in the past, have even broken after 8 months.
Your wageslave task to look for the machine. The springs are actually cheap and so whenever you check, you replace the old spring with a new one, just in case. Now if you check the spring every 9 months, you might mostly be good (the weight keeps on swinging, state A), but it might break (transition to state B) and you lose your job. If instead you check the spring every month you can be pretty sure the system stays in state A.

tl;dr Quantum leaps happen randomly but your lack of knowledge decreases with time (the time being computed back to when you did your last measurement T0). If you measure more frequently (which always resets T0 to a more recent time), you are prone to prevent a leap.

>>7762159
funny thing though (to speak a word of true trivia in this thread), he called it "h" for "Hilfskonstante" in this sense, i.e. "helping constant"

In my notes I consider it a unitless constant pretty consistency, i.e. equate Joule and Hertz and it makes most formulas much easier to read

I think the unsatisfactory comes, in parts, from other mathematicians being autist or yourself being an autist - so you don't connect to the world like some livelong artist would.

>>7397072
As far as I'm concerned, when doing math there is no relevant number beyond 8.

I'm doing my PhD at the German Aerospace center.

I've never been on facebook and the other (female) PhD student I sit in a room with got a 10.000€ girls only stipend.

>>6946879
>Feynman
I think it's kind of overrated. But I might be wrong.
>>6946894
>Srednicki
Those chapters are strangely short, I'm not a fan.
>>6955969
>Weinberg
Then again, if you actually do stuff you must add a calculation book like the above or Peskin and Schröder.
In Weinbergs first book, I especially like the cluster theorem and how he derives all the different possible representations - also the ones where we are generally not interested in a theory about them.
>>6949329
>Greiner
I really like those books too. There are generally surprisingly many german authors which are on the better half of books.
>>6955958
>Wald
I liked the book, but I came to the conclusion that maybe more than other fields, in relativity its tedious to learn the math and the physics at the same time. The math is actually easy, so if you do it first, the physics can be presented more clearly.
At the same time, I've never encountere a good exposition of the equivalence principle.

Has there been a good or approachable textbook on solid state theory.
The standard reference is probably Ashcroft and Mermin, but it's hard when you encounter it first, and later it's not even too good, I think. I don't like Kittel either, or variants.