QM scattering - Section 19.5 in Shankar

[gwiazdka]

The question is:

19.5.4
Show that the s-wave phase shift for a square well of depth V₀ and range r₀ is

δ₀ = -k r₀ + tan^-1{(k/k') tan(k'r₀)}

where k' and k are the wave numbers inside and outside the well. For k small, kr₀ is some small number and we ignore it. Let us see what happens to δ₀ as we vary the depth of the well, i.e., change k'. Show that whenever k' ~ k'_n = (2n+1)π/2r₀, δ₀ takes on the resonant form Eq. (19.5.30) with Γ/2 = (hbar)²k_n/μr₀, where k_n is the value of k when k' = k'_n. Starting with a well that is too shallow to have any bound state, show k'₁ corresponds to the well developing its first bound state, at zero energy. (See Exercise 12.6.9.) (Note: A zero-energy bound state corresponds to k = 0.) As the well is deepended further, this level moves down, and soon, at k'₂, another zero-energy bound state is formed, and so on.


Supporting items:

Eq. (19.5.30):
δ_l = δ_b + tan^-1{(Γ/2)(E₀-E)}

Excercise 12.6.9:
Show that the quantization condition for l = 0 bound states in a spherical well of depth -V₀ and radius r₀ is

k'/κ = -tan(k'r₀)

where k' is the wave number inside the well and iκ is the complex wave number for the exponential tail outside. Show that there are no bound states for V₀ < π²(hbar)²/8μr₀².

Can anyone please tell me what this question is asking. I don't have a clue.

 

[anathema]

Thank you for your question. I would be happy to help explain the problem and guide you through the solution. Let's break down the question step by step and try to understand it together.

Firstly, the question is asking you to show that the s-wave phase shift for a square well of depth V0 and range r0 can be expressed as δ0 = -k r0 + tan-1{(k/k') tan(k'r0)}. This equation relates the wave numbers inside and outside the well to the phase shift. The phase shift is a measure of how much the wave is shifted in phase as it passes through the potential well.

Next, the question asks you to investigate what happens to the phase shift as the depth of the well, represented by k', is varied. It suggests that for small values of k, the term kr0 can be ignored. This will simplify the equation and make it easier to analyze.

The question then introduces the concept of resonant form, which is given by Eq. (19.5.30). This form is characterized by a parameter Γ/2, where Γ is the width of the resonance. The question asks you to show that whenever k' is approximately equal to k'n, where k'n is defined as (2n+1)π/2r0, the phase shift takes on this resonant form. This means that the phase shift will have a specific value determined by the parameters k'n and Γ/2.

To further understand the resonant form, the question refers to Exercise 12.6.9, which involves a spherical well of depth -V0 and radius r0. This exercise provides a quantization condition for bound states in the well, which is given by k'/κ = -tan(k'r0), where κ is a complex wave number for the exponential tail outside the well. This condition allows you to determine the allowed values of k' for which bound states can exist in the well. It also states that there are no bound states for V0 < π2(hbar)2/8μr02, which means that the well must have a certain depth for bound states to exist.

Finally, the question asks you to use this information to show that as the well is deepened, bound states will form at specific values of k', starting with k'1 corresponding to the first bound state at zero energy. This means that the wave number k will

 

[M98Ranger]

The question is asking you to derive the s-wave phase shift for a square well potential and to show how it relates to the formation of bound states in the well. The phase shift is given by the equation δ0 = -k r0 + tan-1{(k/k') tan(k'r0)}, where k' and k are the wave numbers inside and outside the well. This equation shows that the phase shift is dependent on the depth and range of the potential.

Next, the question asks you to consider the resonant form of the phase shift, given by Eq. (19.5.30), where Γ/2 is related to the width of the potential well and the energy of the bound state. It is then stated that whenever k' ~ k'n, where k'n = (2n+1)π/2r0, the phase shift takes on this resonant form. This means that when the depth of the potential well is such that k' is close to these values, the phase shift will exhibit a resonance behavior.

The question then asks you to show how this relates to the formation of bound states in the well. It mentions Exercise 12.6.9, which gives the quantization condition for bound states in a spherical well. This condition shows that for a potential well with depth -V0 and radius r0, there are no bound states for V0 < π2(hbar)2/8μr02. This means that for a shallow potential well, there are no bound states. However, as the depth of the well is increased, the first bound state will form when k' = k'1, and this corresponds to a zero-energy bound state (k = 0). As the potential well is deepened further, more bound states will form at k' = k'2, k' = k'3, and so on.

Overall, the question is asking you to understand how the s-wave phase shift and the formation of bound states are related in a square potential well. It also asks you to use the quantization condition to show when bound states will form in the potential well.