QM: Potential Energy & Box: U(x) & Schrodinger EQ: \psi (x)

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In summary, the potential function U(x) is infinite outside the 1-d rigid box and 0 inside. This is proven by the schrodinger equation when \psi (x) is 0 outside the box.
  • #1
misogynisticfeminist
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1. Why is the potential function U(x) infinite outside the 1-d rigid box and 0 inside? Is this proven by the schrodinger equation when [tex]\psi (x) [/tex] is 0 outside the box?

2. Why is it that in QM, [tex]potential energy= \frac {p^2}{2m}[/tex].

I heard that it is a consequence of the debroglie relations but how can it be if the relations have no mass involved?
 
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  • #2
1.That's the model.The potential is assumeed a constant value inside the box and infite outside.The infinite value of the potential automatically implies the wavefunction to be zero outside the box.

2.You mean KE=p^2/2m...That's CM and the quantization postulate (assuming you mean operators,else it's plain simple CM).

Daniel.
 
  • #3
^ CM?? uhhhh i dun really understand the acronym, can you explain that? thanks alot...And why is the relation a result of the quantization postulate?
 
  • #4
Classical Mechanics...This [tex] \hat{H}=\frac{\hat{p}^{2}}{2m} [/tex] is a result of the 2-nd postulate.

What does that postulate say...?

Daniel.
 
  • #5
hmmm sorry, but i really haven't have a thorough background in CM yet. Is it alright if you explain it here? thanks alot. I'm actually self-taught in QM.
 
  • #6
In short,non mathematized () version,you quantize every classical observable by making the function become a linear operator on the Hilbert space of states...

The momentum & the Hamiltonian are 2 examples...

Daniel.
 
  • #7
misogynisticfeminist said:
hmmm sorry, but i really haven't have a thorough background in CM yet.

Have you at least seen the classical-mechanics formulas for momentum and kinetic energy?

[tex]p = mv[/tex] and [tex]K = \frac {1}{2} mv^2[/tex]

Solve the first equation for v and substitute into the second one.
 
  • #8
I consider myself to be (rather big-headedly) very knowledgeable in QM. But I have to ask, just 'cos [itex]T=\frac{p^2}{2m}[/itex] in CM doesn't mean it should translate directly over into QM, or does it? I know that the observables can be associated with any Hermitian operator, and also that the operators associated with observables should satisfy Heisenberg's Uncertainty relations (namely [itex]\mathbf{xp-px}=i\hbar[/itex]).

So once we have chosen the operator for position to be pre-multiply by the position, the moment operator follows, since

[tex](\mathbf{x})(-i\hbar\nabla)\psi-(-i\hbar\nabla)(\mathbf{x})\psi=i\hbar\psi[/tex]

as required. So I'm happy with the operators for position and momentum. But I question the logic in choosing the operators for energy and angular momentum based on their classical definitions. (I'm not saying they're wrong because I know they work remarkably well).

My misunderstanding may well arise because (as we were discussing earlier) I only know the Lagrangian formalism in detail, and also QM in detail, but do not yet know the Hamiltonian formalism in detail.

In fact, this document shows how I learned QM: http://users.ox.ac.uk/~quee1685/main.pdf . I would appreciate if a few people (in particular dextercioby and zapperz) could provide criticism on it.
 
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  • #9
masudr said:
I consider myself to be (rather big-headedly) very knowledgeable in QM. But I have to ask, just 'cos [itex]T=\frac{p^2}{2m}[/itex] in CM doesn't mean it should translate directly over into QM, or does it?

There are rules.The postulate of quantization explains them very well.U may search for "Weyl ordering".

masudr said:
I know that the observables can be associated with any Hermitian operator, and also that the operators associated with observables should satisfy Heisenberg's Uncertainty relations (namely [itex]\mathbf{xp-px}=i\hbar[/itex]).

No,that's a particular case of a far more general statement

masudr said:
So once we have chosen the operator for position to be pre-multiply by the position, the moment operator follows, since

[tex](\mathbf{x})(-i\hbar\nabla)\psi-(-i\hbar\nabla)(\mathbf{x})\psi=i\hbar\psi[/tex]

as required. So I'm happy with the operators for position and momentum.

There's much more to it.The proof for [tex] \langle \vec{r}|\hat{\vec{P}}|\psi\rangle =-i\hbar\nabla_{\vec{r}}\psi(\vec{r}) [/tex] is rather tedious...

masudr said:
But I question the logic in choosing the operators for energy and angular momentum based on their classical definitions.(I'm not saying they're wrong because I know they work remarkably well).

Then u should read either Roger P.Newton's latest book on QM ("Quantum Theory:A Text for Graduate Students",Springer Verlag,2002) (i'm sure you've heard of him,he's an Englishman),or J.J.Sakurai's masterpiece.They take nothing for granted and they don't use the "traditional" axiomatic approach to (non-relativistic) QM in Dirac's formulation.

masudr said:
My misunderstanding may well arise because (as we were discussing earlier) I only know the Lagrangian formalism in detail, and also QM in detail, but do not yet know the Hamiltonian formalism in detail.

As i said,u should... :wink: If you're really interested in an overview of this theory...

masudr said:
In fact, this document shows how I learned QM: http://users.ox.ac.uk/~quee1685/main.pdf . I would appreciate if a few people (in particular dextercioby and zapperz) could provide criticism on it.

Rushed,missed a few key points in the axioms (which is very bad),introduced (for the purpose of the article) unnecessary mathematical details...

Useless...Better put a hand (actually both) on David J.Griffiths' (another Englishman) book.

Daniel.
 
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  • #10
dextercioby said:
There are rules.The postulate of quantization explains them very well.U may search for "Weyl ordering".

I know Weyl ordering; it makes sense now.

No,that's a particular case of a far more general statement

I am aware that HUP applies to any pair of observables whose operators don't commute. Is there a more general statement I'm missing?

There's much more to it.The proof for [itex] \langle \vec{r}|\hat{\vec{P}}|\psi\rangle =-i\hbar\nabla_{\vec{r}}\psi(\vec{r}) [/itex] is rather tedious...

OK. I can accept that.

Rushed,missed a few key points in the axioms (which is very bad),introduced (for the purpose of the article) unnecessary mathematical details...

Yes it was rushed! I have, funnily enough, better things to do than summarise my knowledge on QM! Although I admit that's no excuse for sloppiness. Care to elaborate on which axioms I missed out? By unnecessary mathematics, I assume you mean groups/fields? This was the way I was introduced to vector spaces; besides it just delegates listing all the axioms of a Hilbert space to previous sections; and gives the reader glimpses into other fields of mathematics.

You have recommended many books to me; I will certainly take a look.

Masud.
 
  • #11
masudr said:
I am aware that HUP applies to any pair of observables whose operators don't commute. Is there a more general statement I'm missing?

Yes,any QM book has a jusitification for the general relation which applies to all possible pairs of QM observables.


masudr said:
By unnecessary mathematics, I assume you mean groups/fields? This was the way I was introduced to vector spaces;

At least that should have been the prerequisite in maths one might have...


masudr said:
You have recommended many books to me; I will certainly take a look.

Masud.

If you have the time & the will...

Daniel.

EDIT:And one more thing,Masud,HAPPY BIRTHDAY!
 
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  • #12
dextercioby said:
U may search for "Weyl ordering".

I tried googling on Weyl ordering, and I found this excellent tutorial on Deformation Quantizing: http://idefix.physik.uni-freiburg.de/~stefan/Skripte/intro/node1.html
 
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FAQ: QM: Potential Energy & Box: U(x) & Schrodinger EQ: \psi (x)

What is potential energy in quantum mechanics?

Potential energy in quantum mechanics refers to the energy that is associated with the position of a particle in a system. It is a function of the position of the particle and can be described by the potential energy function, U(x). This potential energy can affect the behavior and properties of the particle as described by the Schrodinger equation.

What is the significance of the potential energy function, U(x), in quantum mechanics?

The potential energy function, U(x), plays a crucial role in determining the energy levels and properties of a quantum system. It describes the potential energy of a particle in a given position and can help predict the behavior and stability of the system. The shape and behavior of the potential energy function can also provide insight into the interactions between particles in the system.

How is the Schrodinger equation related to potential energy in quantum mechanics?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior and properties of a quantum system. It takes into account the potential energy function, U(x), in its formulation and helps determine the energy levels and wave function of the system. The Schrodinger equation is a powerful tool for understanding the relationship between potential energy and the behavior of particles in a quantum system.

Can potential energy in quantum mechanics be negative?

Yes, potential energy in quantum mechanics can be negative. This is because potential energy is a relative quantity and is defined as the difference in energy between two states. Therefore, the potential energy can take on both positive and negative values depending on the reference point. In some cases, a negative potential energy can indicate a bound state, where the particle is confined to a certain region.

How does the box potential, U(x), affect the behavior of a particle in a quantum system?

The box potential, U(x), refers to a potential energy function that is constant within a certain region (the box) and infinite outside of it. This potential energy function creates a finite potential well for the particle, which can lead to quantized energy levels and confinement of the particle within the box. The box potential is often used as a simplified model for studying the behavior of particles in a confined space.

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