Minimum value of V_0 for Odd Solutions in Square Well Potential

In summary, we discussed the bound states of a particle in a square well potential with a given depth. We found that the even solutions have energies that satisfy a transcendental equation, while the odd solutions have energies that satisfy a different transcendental equation. We also explored the relationship between the real and complex wave numbers inside and outside the well, and how they are related by a constant determined by the depth of the well. Additionally, we examined a graphical representation of the bound states in the (##\alpha = ka, \beta = \kappa a ##) plane, where the bound states are given by the intersection of the curve ##\alpha \cot \alpha = -\beta## with a circle. Finally, we determined that there is
  • #1
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Homework Statement


Consider a particle in a square well potential:

$$V(x) = \begin{cases} 0, & |x| \leq a \\ V_0, & |x| \geq a \end{cases} $$

We are interested in the bound states i.e. when ##E \leq V_0##.

(1) Show that the even solutions have energies that satisfy the transcendental equation $$k \tan ka = \kappa$$ while the odd ones will have energies that satisfy $$k \cot ka = -\kappa$$ where ##k## and ##i\kappa## are real and complex wave numbers inside and outside the well respectively. Note that ##k## and ##\kappa## are related by $$k^2 + \kappa^2 = 2mV_0/\hbar^2$$

(2) In the (##\alpha = ka, \beta = \kappa a ##) plane, imagine a circle that obeys the above. The bound states are then given by the intersection of the curve ##\alpha = \tan \alpha = \beta ## or ## \alpha \cot \alpha = -\beta ## with the circle.

(3) Show that there is always one even solution and that there is no odd solution unless ## V_0 \geq \hbar^2 \pi^2 /8ma^2. ## What is ##E## when ##V_0## just meets this requirement?

Homework Equations

The Attempt at a Solution



I was able to solve part (1) by applying the boundary conditions on ##e^{-\kappa|x|}## for ##|x| \geq a## and ##e^{ikx} + e^{-ikx}## and ##e^{ikx} - e^{-ikx}##.

I 'sort' of understand why the solutions wouldn't always exist for part (3) after plotting the curve ##\alpha \cot \alpha## as it is only positive, and we are looking for solutions that are negative. I'm not quite sure how to find the minimum value of ##V_0##.
 
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  • #2
The relevant equation (TISE) should be shown.

The ground state is the even solution.

The first odd solution is the 1st excited state, which requires a minimum depth to the well.
 
  • #3
I assumed I should use the relations which came about from the TISE?

I realized I plotted the incorrect graphs. If the root of ##\alpha \cot \alpha## is at ##\pi/2##, then that is the minimum value of the radius. Thus for ##k## and ##\kappa## it is ##\pi/2a##, which means that ##2mV_0/\hbar^2 = \pi^2/4a^2##.
 
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FAQ: Minimum value of V_0 for Odd Solutions in Square Well Potential

1. What is a square well potential?

A square well potential is a commonly used model in physics to describe the potential energy of a particle in a confined space. It is a square-shaped potential energy function that is constant inside the well and infinite outside the well.

2. How is a square well potential created?

A square well potential can be created by placing a physical barrier, such as a wall or a box, around a certain region in space. This creates a confined space where the particle can only exist within the boundaries of the well.

3. What are the properties of a square well potential?

A square well potential has two main properties: depth and width. The depth refers to the potential energy inside the well, while the width refers to the size of the well. These properties can be adjusted to simulate different physical systems.

4. What are the applications of a square well potential?

A square well potential has various applications in physics, such as in quantum mechanics, solid state physics, and nuclear physics. It is commonly used as a simple model to understand the behavior of particles in confined spaces.

5. What are the limitations of a square well potential?

A square well potential is a simplified model and does not accurately represent all physical systems. It assumes that the potential energy is constant inside the well, which may not always be the case. Additionally, it does not take into account the effects of external forces on the particle.

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