What is the condition for a particle to go into a square potential well?

[learner@123]

Hello everyone

I am searching for the answer for the condition (related to the total energy of the particle E) for which any particle will go into the square potential well.
I have studied Griffiths's quantum mechanics book Introduction to the quantum mechanics Section 2.5 and 2.6) but still not able to relate this to practical situation .
See the attachment for book.

Will be very thankful to your answers
Thanks .

 

[BvU]

Hi there, and welcome to PF !

I am afraid your question isn't completely clear to me: what exactly do you mean with

go into the square potential

since the section should in fact make clear that this isn't what's happening in quantum mechanics ...

And: I assume that you are referring to a finite well (section 2.6) ?

 

[learner@123]

Thanks for ur positive response

Actually sir i have confusion on trap condition and scattering condition for electron in finite potential well.
In given section 2.6 author discussed two different cases (1)when total energy E is negative i.e trapping condition (2)when total energy E is positive i.e scattering condition.
Now if any electron is traveling on vacuum level its total energy(=PE+KE ) will be always positive since it will always have some kinetic energy that means it will never get into the well.
So sir my confusion is in which condition it will go into the well.

Thanks

 

[Nugatory]

Now if any electron is traveling on vacuum level its total energy(=PE+KE ) will be always positive since it will always have some kinetic energy that means it will never get into the well.

The potential energy can be negative, so there's no reason why the sum (PE+KE) cannot be less than zero even though the kinetic energy is always positive.

 

[BvU]

Dear learner,

Your "if any electron is travelling... " is about positive energies; for such particles there is a scattering process going on at the position of the well. The solution of the ( for such a case time dependent !) Schroedinger equation is pretty involved. I think there exist animations on the net but I don't know offhand where. http://physics.oregonstate.edu/~rubin/nacphy/CPapplets/QMWAVE/Potbar/Potbar.html?

Anyway, the picture of "falling into the well" isn't very realistic -- in a simple case there is nowhere to go for the energy, so it doesn't stay in if it starts outside.

 

[jtbell]

in a simple case there is nowhere to go for the energy, so it doesn't stay in if it starts outside.

Exactly. For the idealized situation with the incoming electron represented by a plane wave (definite momentum and kinetic energy), you can calculate transmission and reflection probabilities using the method shown on pages 81-82 of the PDF in the first post. However, those always add to give a total probability of 1. If you want the electron to be trapped inside the well (become a bound state), then you have to have a mechanism for changing the energy from a value > 0 to a value < 0.

 

[learner@123]

The potential energy can be negative, so there's no reason why the sum (PE+KE) cannot be less than zero even though the kinetic energy is always positive.

Sir actually I mentioned that electron is at vacuum level hence its PE will be zero but it will have some KE.

 

[learner@123]

Dear learner,

Your "if any electron is travelling... " is about positive energies; for such particles there is a scattering process going on at the position of the well. The solution of the ( for such a case time dependent !) Schroedinger equation is pretty involved. I think there exist animations on the net but I don't know offhand where. http://physics.oregonstate.edu/~rubin/nacphy/CPapplets/QMWAVE/Potbar/Potbar.html?

Anyway, the picture of "falling into the well" isn't very realistic -- in a simple case there is nowhere to go for the energy, so it doesn't stay in if it starts outside.

Sir can you suggest any literature regarding that logic it will be good for me ...

 

[Nugatory]

Sir actually I mentioned that electron is at vacuum level hence its PE will be zero but it will have some KE.

Ah - yes, if you are dealing with a particle that is approaching from infinity, then it can only end up in a bound state if it can shed some of its energy in some other interaction so that it can be captured by the potential well.

Although you came across this problem in a Quantum Mechanics text, the same issues arise in classical mechanics. We cheerfully pose classical problems involving bound particles (planetary orbits, for example) without considering how they came to be bound in the first place. Obviously they do, as the universe is full of particles that are in bound states; often the mechanism by which this happened is irrelevant to the problem at hand.

 

[learner@123]

Exactly. For the idealized situation with the incoming electron represented by a plane wave (definite momentum and kinetic energy), you can calculate transmission and reflection probabilities using the method shown on pages 81-82 of the PDF in the first post. However, those always add to give a total probability of 1. If you want the electron to be trapped inside the well (become a bound state), then you have to have a mechanism for changing the energy from a value > 0 to a value < 0.

Sir you got my problem exactly as i was thinking but still if suppose there are certain energy levels just below vacuum level then Is it possible for electron to loose its kinetic energy to go into lower energy level .How energy conservation law holds

 

[learner@123]

Ah - yes, if you are dealing with a particle that is approaching from infinity, then it can only end up in a bound state if it can shed some of its energy in some other interaction so that it can be captured by the potential well.

Although you came across this problem in a Quantum Mechanics text, the same issues arise in classical mechanics. We cheerfully pose classical problems involving bound particles (planetary orbits, for example) without considering how they came to be bound in the first place. Obviously they do, as the universe is full of particles that are in bound states; often the mechanism by which this happened is irrelevant to the problem at hand.

thank you sir for ur quick reply ...now i understood sir...but sir if the particle is not coming from infinity then what will be the situation on trapping

 

[BvU]

A careful word of advice: while still in chapter 2 of Griffiths, don't attempt to make direct links from the material presented to the daily real world. Quantum Mechanics lives in its own miraculous environment and you first want to acquaint yourself with its tools and its methods while studying isolated and exotic phenomena. That's already quite a huge task in itself.

 

[jtbell]

Sir you got my problem exactly

A friendly tip: you don't need to call us "sir."