Schrodinger equotion & quantum physics

[dreamfly]

i'm a green hand in this field,maybe the question i asked is droll,but i want to know some wonder thing about quantum physics.
i know that one dimensional potential trap can be reckoned from Schrödinger equotion, but what about the actually fact? How dose one dimensional potential trap form? & Why did Schrödinger bring the equotion forward in that form?

Thanks!

 

[seratend]

i'm a green hand in this field,maybe the question i asked is droll,but i want to know some wonder thing about quantum physics.
i know that one dimensional potential trap can be reckoned from Schrödinger equotion, but what about the actually fact? How dose one dimensional potential trap form? & Why did Schrödinger bring the equotion forward in that form?
Thanks!

When you write the hamiltonian H= p^2/2m + V(x) for a particle:
H|psi>=ihhard/dt|psi>, where |psi> is the state of the particle, you should take into account the state of the system that creates the V(x) potential (hidden in the above simplified equation and not important for the results.

Let's call the state of the system potential |potential> and assume it is an eigen state of the free hamiltonian (Ho_potential) of the potential system so that the |potential> state remains through the unitary evolution.

We have the complete hamiltonian given by:

H=p^2/2m + |potential><potential|.V(x)+Ho_potential

Where the complete state is |psi(t)>|potential> for the particle and the potential system.

=> <potential|H|psi(t)>|potential>=
[p^2/2m + V(x)]|psi(t)>+<potential|Ho_potential|potential>|psi>= [p^2/2m+V(x)+cte]|psi>
= ihbar d/dt<potential|psi>|potential>= ihbar d/dt|psi>

=> we have recoverd the unitary evolution of the state |psi(t)> (H is defined up to a constant.


Seratend.

 

[dextercioby]

There's quite a long story on how Schrödinger coined his equation.He really believed in that optics analogy and describing the atom through waves similar to the ones proposed by Louis de Broglie in his PhD thesis in Nov.1924.

Bottom line,he found it and applied it to the H atom.Then it was unanimously accepted and incorporated by Dirac in his formalism.

Daniel.

 

[dreamfly]

Well,it's seems that we can detect many facts from equotions,so i preciate how marvellous the universe is.But why & how position and velocity become uncertain when objects become smaller and smaller?Is it decided by objects' time-space property?

 

[selfAdjoint]

Well,it's seems that we can detect many facts from equotions,so i preciate how marvellous the universe is.But why & how position and velocity become uncertain when objects become smaller and smaller?Is it decided by objects' time-space property?

It's due to the fact that the two observations, of position and of momentum, can't be done simultaneously, and that is because the order in which they are made makes a difference. The mathematical jargon for this is that they fail to commute. Commutation rules are required for quantization, so we can say that uncertainty comes deeply out of the quantum nature of the world.

 

[dextercioby]

Incidentally,at classical level,general coordinates & canonical momenta fail to (anti) commute in the (graded) Poisson brackett.

So assuming Dirac's quantization scheme

graded Poisson brackett goes to $$\frac{1}{i\hbar}$$ times graded Lie brackett,it all makes perfect sense.

Daniel.

 

[dreamfly]

and that is because the order in which they are made makes a difference. Commutation rules are required for quantization, so we can say that uncertainty comes deeply out of the quantum nature of the world.

Poisson brackett.

But i still have some puzzles:what's"the order in which they are made"?and what's "Poisson brackett"?and does it mean that the quantum nature is a nature of the world?but how unimaginable it is!will the God dicing,really?

 

[dextercioby]

I'm sorry,but if you don't know about Poisson brackets,then u should you go back to school and learn...

And also ask them about "incompatible obserables" in QM.

Daniel.

 

[dreamfly]

I'm sorry,but if you don't know about Poisson brackets,then u should you go back to school and learn...

And also ask them about "incompatible obserables" in QM.

Daniel.

yes I'm in university now.but we haven't study it normally & deeply.i only know some basic knowledge about quantum physics,like uncertainty principle,Schrödinger equotion...according in being knowledge, i got some basic puzzles. and want to get the answers here.so i beg your instruction and thanks again for answering my droll questions!

 

[James Jackson]

To save me attempting to explain, this page seems to give a good intro:

http://farside.ph.utexas.edu/teaching/qm/fundamental/node22.html

I haven't read it in detail, with the excuse being I finished my degree this morning...

 

[oklobiaochai]

please send me the Schrödinger time dependent wave equation and also diagrams describing potential well

 

[James Jackson]

Wave equation is, as always, $\hat H |\Psi\rangle = E|\Psi\rangle$. The Hamiltonian $\hat H$ expresses the time dependance, and is dependent on the system - you can't just give a generic time dependent Hamiltonian.

I suggest you start more simply before looking at time dependance...