Delta-function potentials, scattering states

In summary, the Introduction to Quantum Mechanics book by Griffiths discusses delta-function potential wells and the general solution for finding bound states with E < 0 and E > 0, x < 0. The solution for E > 0 does not have a term that blows up as x goes to -∞.
  • #1
AntiStrange
20
1
I'm reading through the Introduction to Quantum Mechanics book by Griffiths (2nd edition)
and it is describing delta-function potential wells.
When it describes how to find bound states the energy is E < 0 (negative) in the region x < 0 (negative).

It says the general solution is:
[tex]\psi (x) = Ae^{-\kappa x} + Be^{\kappa x}[/tex]
It explains that as x goes to -∞ the first term blows up. Which makes sense because it will go to infinity, while the second term only goes to zero.

However, they then look at the situation when E > 0, x < 0
again the general solution is:
[tex]\psi (x) = Ae^{-\kappa x} + Be^{\kappa x}[/tex]

but this time they say that neither term blows up. I don't understand why that is. Shouldn't the first term still go to infinity as x goes to -∞ ?
 
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  • #2
No, for E > 0, x < 0 the general solution is
[tex]
\psi(x) = Ae^{-ikx} + Be^{ikx}
[/tex]
which oscillates but does not exponentially grow or decay (for real k).
 
  • #3


I can provide a response to the content regarding delta-function potentials and scattering states. The delta-function potential, also known as the Dirac delta potential, is a mathematical function used to model a potential well in quantum mechanics. It is represented by a spike at a specific point in space, with an infinitely large value at that point and zero value everywhere else.

In the context of scattering states, the delta-function potential can be used to study the behavior of particles as they interact with the potential well. In the case of bound states, where the energy is less than zero, the general solution for the wave function involves two terms - one that exponentially increases as x approaches -∞ and one that exponentially decreases. This makes sense, as the particle is confined to the potential well and cannot escape.

However, when the energy is greater than zero, the particle is no longer bound to the potential well and can escape. In this case, the general solution still involves two terms, but both terms decrease exponentially as x approaches -∞. This may seem counterintuitive, as one might expect the first term to blow up as x approaches -∞. However, this is not the case because the delta-function potential is not a physical potential and does not actually exist at x = -∞. Therefore, the wave function does not blow up at this point and remains finite.

In summary, the behavior of the wave function in the presence of a delta-function potential depends on the energy of the particle. In the case of bound states, the wave function exponentially increases as x approaches -∞, while in the case of scattering states, both terms in the general solution decrease exponentially. This is due to the nature of the delta-function potential and its non-existence at x = -∞.
 

1. What is a delta-function potential?

A delta-function potential is a mathematical construct used in quantum mechanics to represent a sudden and localized change in the potential energy of a particle. It is often used to model interactions between particles or with a potential barrier.

2. How does a delta-function potential affect scattering states?

A delta-function potential can cause scattering states in quantum systems, where a particle interacts with the potential for a short period of time before moving on. The potential can change the direction and energy of the particle, leading to scattering and interference effects.

3. What are the properties of scattering states in a delta-function potential?

Scattering states in a delta-function potential have a continuous energy spectrum, meaning that there are an infinite number of possible energy levels. They also have a discrete set of scattering angles, which depend on the strength and location of the potential.

4. How does the strength of a delta-function potential affect scattering states?

The strength of a delta-function potential can greatly influence scattering states. A stronger potential will cause more significant changes in the particle's energy and direction, while a weaker potential will have less of an effect. The location of the potential also plays a role in determining the scattering behavior.

5. What are some real-life applications of delta-function potentials and scattering states?

Delta-function potentials and scattering states have many practical applications in physics and engineering. They are used to model interactions between particles in nuclear physics, quantum field theory, and solid state physics. They are also used in the study of quantum scattering experiments and in the design of electronic devices such as transistors and sensors.

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