Finite Square Well: Constructing Initial Wave Function with E < 0

In summary, the conversation discusses the derivation of bound and scattering states for a finite square well. The speaker understands the logic behind it but is unsure how to accommodate an arbitrary initial wave function with a mean energy less than zero. They suggest that the initial state could be a unique mixture of bound and scattering states, but question how this could result in an energy lower than the lowest bound state. They also consider the possibility that such particles may not exist. However, the speaker eventually figures out the solution in terms of uncertainty principles.
  • #1
genxhis
37
1
I read through the derivation of bound and scattering states for a finite square well. The logic made sense to me, but I am not entirely sure how to accommodate an arbitrary initial wave function (with mean E < 0). Afterall, there are only a finite number of bound states. My guess was that the initial state might be constructed as a (unique?) mixture of both bound and scattering states. But how could this build a state with energy less than that of the lowest bound state?

Could someone clarify? thx.
 
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  • #2
Maybe such particles cannot exist (?)

edit: okay, i think i have it figured out in terms of uncertainty principles.
 
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  • #3


Your guess is correct, in order to construct an initial wave function with an energy less than the lowest bound state, we need to consider a mixture of both bound and scattering states. This is because the bound states represent the discrete energy levels available in the finite square well, while the scattering states represent the continuous energy spectrum. By combining these two types of states, we can create an initial wave function with any energy, including those below the lowest bound state.

To understand this concept better, let's consider the time-independent Schrodinger equation for a particle in a finite square well:

$\frac{d^2\psi}{dx^2} = \frac{2m}{\hbar^2}(V_0 - E)\psi$

where $V_0$ is the depth of the well and $E$ is the energy of the particle. This equation has two types of solutions: bound states and scattering states. Bound states have discrete energy levels given by:

$E_n = \frac{\hbar^2k_n^2}{2m}$

where $k_n = \frac{n\pi}{L}$ and $n$ is a positive integer. These bound states are represented by the wave functions:

$\psi_n(x) = A_n\sin(k_nx)$

where $A_n$ is a normalization constant.

On the other hand, scattering states have a continuous energy spectrum and are represented by the wave functions:

$\psi_k(x) = B_ke^{ikx} + C_ke^{-ikx}$

where $k$ is the wave number and $B_k$ and $C_k$ are normalization constants.

Now, to construct an initial wave function with energy less than the lowest bound state, we can take a linear combination of both bound and scattering states:

$\psi(x) = \sum_{n=1}^{\infty}a_n\psi_n(x) + \int_{0}^{\infty}b_k\psi_k(x)dk$

where $a_n$ and $b_k$ are coefficients that determine the weight of each state in the linear combination. By choosing the appropriate values for these coefficients, we can create an initial wave function with any energy, including those below the lowest bound state.

In summary, the key idea is to combine both bound and scattering states to create an initial wave function with the desired energy. This approach is valid because any wave function can be represented as a
 

Related to Finite Square Well: Constructing Initial Wave Function with E < 0

1. What is a finite square well?

A finite square well is a potential energy barrier that has a finite depth and width. It is often used as a simplified model for a particle moving in a confined space or a bound state.

2. How is the initial wave function constructed for a particle in a finite square well with negative energy?

The initial wave function for a particle in a finite square well with negative energy is constructed by using the Schrödinger equation and boundary conditions. The wave function must be continuous and differentiable at the boundaries of the well, and the solution must satisfy the energy equation for a negative energy state.

3. What is the significance of having a negative energy state in a finite square well?

A negative energy state in a finite square well indicates that the particle is bound within the well. This means that the particle has a finite amount of energy and cannot escape the well without external energy being applied.

4. How does the width and depth of the finite square well affect the initial wave function for a particle with negative energy?

The width and depth of the finite square well directly affect the shape and amplitude of the initial wave function. A wider and deeper well will result in a lower energy state and a more confined and localized wave function. A narrower and shallower well will lead to a higher energy state and a more spread out wave function.

5. Can a particle with negative energy in a finite square well tunnel through the potential barrier?

Yes, a particle with negative energy in a finite square well can tunnel through the potential barrier if the barrier is not infinitely high. This is a quantum phenomenon where the particle can have a non-zero probability of being found outside of the well, even though it does not have enough energy to overcome the barrier classically.

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