QM Infinite square well with delta function potential in middle

In summary, the delta function potential in the infinite square well represents a narrow and strong barrier that divides the well into two halves. It affects the energy levels by causing them to split and shift, and can be solved analytically using perturbation theory. The wave function in its presence is continuous but its derivative is discontinuous, resulting in a kink. The physical interpretation is that it represents a barrier or boundary in the system and has implications for the particle's probability distribution.
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The wave function itself will be continuous across x=0, but its derivatives will not.
You will not really get two separate wavefunctions, but you can try solving the DE in the respective regions and joining them later.
 

FAQ: QM Infinite square well with delta function potential in middle

1. What is the significance of the delta function potential in the infinite square well?

The delta function potential represents an infinitely narrow and infinitely strong barrier at a specific point in the potential energy function. In the case of the infinite square well, this represents a barrier in the center of the well, dividing it into two halves.

2. How does the delta function potential affect the energy levels in the infinite square well?

The presence of the delta function potential alters the energy levels in the infinite square well, causing them to shift and split into two sets of energy levels. This is due to the fact that the delta function potential acts as a perturbation in the Hamiltonian of the system.

3. Can the delta function potential be solved analytically in the infinite square well?

Yes, the delta function potential in the infinite square well can be solved analytically using the method of perturbation theory. This involves treating the delta function potential as a small perturbation and using the first-order correction to the energy levels.

4. How does the wave function behave in the presence of the delta function potential?

The wave function in the infinite square well with a delta function potential is continuous at the point of the potential barrier, but its derivative is discontinuous. This results in a kink or jump in the wave function at the point of the potential barrier.

5. What is the physical interpretation of the delta function potential in the infinite square well?

The physical interpretation of the delta function potential in the infinite square well is that it represents a barrier or boundary in the system. This could be a physical barrier, such as a thin wall, or an abstract boundary, such as a change in the potential energy function. It also has implications for the probability of finding a particle in a particular region of the well.

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