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A particle starts out (at time t=0 ) in the Nth state of the infinite square well. Now the 鈥渇loor鈥� of the well rises temporarily (maybe water leaks in, and then drains out again), so that the potential inside is uniform but time dependent:${\mathit{V}}_{\mathbf{0}}\left(t\right)\mathbf{,}\mathbf{}\mathbf{with}\mathbf{}{\mathit{V}}_{\mathbf{0}}\left(0\right)\mathbf{=}{\mathit{V}}_{\mathbf{0}}\left(T\right)\mathbf{=}\mathbf{0}$.

(a) Solve for the exact ${\mathit{c}}_{\mathbf{m}}\left(t\right)$, using Equation 11.116, and show that the wave function changes phase, but no transitions occur. Find the phase change, role="math" localid="1658378247097" $\mathbf{蠒}\mathbf{\left(}\mathbf{T}\mathbf{\right)}$, in terms of the function ${\mathit{V}}_{\mathbf{0}}\left(t\right)$

(b) Analyze the same problem in first-order perturbation theory, and compare your answers. Compare your answers.
Comment: The same result holds whenever the perturbation simply adds a constant (constant in x, that is, not in to the potential; it has nothing to do with the infinite square well, as such. Compare Problem 1.8.

Step by step solution

01

Step 1: First-Order perturbation theory

The first-Order perturbation equation includes all the terms in the Schrodinger equation $H_{蠄}=E_{蠄}$that represent the first order approximations to $H_{,}{蠄}_{,}{E}_{,}$This equation can be obtained by truncating $H_{,}{蠄}_{,}{E}_{,}$after the first order terms.

02

Step 2: (a) Solving for 

Equation 9.82

${\stackrel{藱}{c}}_m=-\frac{i}{魔}\underset{n}{鈭�}{c}_n{H^{\text{'}}}_{mn}{e}^i\left(E_m-E_n\right)t/魔$

Here,

鈥︹��. (11.116)

.Let,

$$\begin{gathered} 桅\left(t\right)=-\frac{1}{魔}{鈭�}_0^t{V}_0\left(t^{\text{'}}\right)dt; \\ {c}_m\left(t\right)=e^{i桅}{c}_m\left(0\right) \end{gathered}$$

Hence,

${\left|c_m\left(t\right)\right|}^2={\left|c_m\left(0\right)\right|}^2,\mathrm{and}\mathrm{there}\mathrm{are}\mathrm{no}\mathrm{transitions}.桅\left(T\right)=-\frac{1}{魔}{鈭�}_0^T{V}_0\left(t\right)dt$

03

Step 3: (b) Analyzing the problem in the first order perturbation theory

$$\begin{gathered} c_N\left(t\right)鈮�1-\frac{i}{魔}{鈭�}_0^t{V}_0\left(t^{\text{'}}\right)dt \\ =1+i桅 \\ {c}_m\left(t\right)=-\frac{i}{魔}{鈭�}_0^t{未}_{mN}{V}_0\left(t^{\text{'}}\right)e^{i\left(E_m-E_N\right)t^{\text{'}}/魔}dt \\ =0\left(m鈮�N\right) \\ {c}_m\left(t\right)鈮�-\frac{i}{魔}{鈭�}_0^t{H^{\text{'}}}_{mN}\left(t^{\text{'}}\right)e^{i\left(E_m-E_N\right)t^{\text{'}}/魔}d{t}^{\text{'}},(m鈮�N) \end{gathered}$$

The exact answer is $c_N\left(t\right)=e^{i桅\left(t\right)},c_m\left(t\right)=0$, and they are consistent since, $e^{i桅}鈮�1+i$ to first order.

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