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A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states:

$\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{=}\mathbf{A}\mathbf{\left[}{\mathbf{蠄}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}{\mathbf{蠄}}_{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right]}$

You can look up the series

$\frac{\mathbf{1}}{{\mathbf{1}}^{\mathbf{6}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{3}}^{\mathbf{6}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{5}}^{\mathbf{6}}}\mathbf{+}\mathbf{鈰�}\mathbf{=}\frac{{\mathbf{蟺}}^{\mathbf{6}}}{\mathbf{960}}$

and

$\frac{\mathbf{1}}{{\mathbf{1}}^{\mathbf{4}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{3}}^{\mathbf{4}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{5}}^{\mathbf{4}}}\mathbf{+}\mathbf{鈰�}\mathbf{=}\frac{{\mathbf{蟺}}^{\mathbf{4}}}{\mathbf{96}}$

in math tables. under "Sums of Reciprocal Powers" or "Riemann Zeta Function."

(a) Normalize $\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{)}$. (That is, find A. This is very easy, if you exploit the orthonormality of ${\mathbf{蠄}}_{\mathbf{1}}$ and ${\mathbf{蠄}}_{\mathbf{2}}$. Recall that, having ${\mathbf{蠄}}_{}$normalized at , $\mathbf{t}\mathbf{=}\mathbf{0}$ , you can rest assured that is stays normalized鈥攊f you doubt this, check it explicitly after doing part(b).

(b) Find$\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$ and${\mathbf{\left|}\mathbf{唯}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{\right|}}^{\mathbf{2}}$ . Express the latter as a sinusoidal function of time. To simplify the result, let$\mathbf{蝇}\mathbf{鈮�}\raisebox{1ex}{${\mathbf{蟺}}^{\mathbf{2}}\mathbf{鈩�}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}{\mathbf{ma}}^{\mathbf{2}}$}\right.$

(c)Compute $\mathbf{鉄�}\mathbf{x}\mathbf{鉄�}$ . Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation?(If your amplitude is greater than $\raisebox{1ex}{$\mathbf{a}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$, go directly to jail.

(d) Compute $\mathbf{鉄�}\mathbf{p}\mathbf{鉄�}$.

(e) If you measured the energy of this particle, what values might you get, and what is the probability of getting each of them? Find the expectation value of$\mathbf{H}$ . How does it compare with E₁ and E₂

Although the overall phase constant of the wave function is of no physical significance (it cancels out whenever you calculate a measurable quantity), the relative phase of the coefficients in Equation 2.17 does matter. For example, suppose we change the relative phase of ${\mathit{蠄}}_{\mathbf{1}}\mathbf{}\mathbf{and}\mathbf{}{\mathit{蠄}}_{\mathbf{2}}$in problem 2.5:$\mathit{蠄}\left(x,0\right)\mathbf{=}\mathit{A}\left[{蠄}_1\left(x\right)+e^{i蠒}{蠄}_2\left(x\right)\right]$Where $\mathit{蠒}$is some constant. Find $\mathit{蠄}\left(x,t\right)\mathbf{,}\mathbf{}{\left|蠄\left(x,t\right)\right|}^{\mathbf{2}}$, and $\left(x\right)$, and compare your results with what you got before. Study the special cases $\mathit{蠒}\mathbf{=}\raisebox{1ex}{$\mathbf{蟺}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\mathbf{}\mathbf{and}\mathbf{}\mathit{蠒}\mathbf{=}\mathbf{蟺}$.

A particle in the infinite square well has the initial wave function

$\mathit{蠄}\left(X,0\right)\mathbf{=}\{\begin{array}{l}Ax,0鈮�x鈮�\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ A\left(a-x\right),\raisebox{1ex}{$a$}\!\left/ \!\raisebox{-1ex}{$2$}\right.鈮�x鈮�a\end{array}$

(a) Sketch $\mathbf{蠄}\left(x,0\right)$, and determine the constant A

(b) Find$\mathbf{蠄}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}$

(c) What is the probability that a measurement of the energy would yield the value${\mathit{E}}_{\mathit{1}}$ ?

(d) Find the expectation value of the energy.

A particle of mass m in the infinite square well (of width a) starts out in the left half of the well, and is (at $\mathbf{t}\mathbf{=}\mathbf{0}$) equally likely to be found at any point in that region

(a) What is its initial wave function, $\mathbf{蠄}\left(x,0\right)$? (Assume it is real. Don鈥檛 forget to normalize it.)

(b) What is the probability that a measurement of the energy would yield the values$\raisebox{1ex}{${\mathbf{蟺}}^{\mathbf{2}}{\sout{\mathbf{h}}}^{\mathbf{2}}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}{\mathbf{ma}}^{\mathbf{2}}$}\right.$?

For the wave function in Example 2.2, find the expectation value of H, at time $\mathbf{t}\mathbf{=}\mathbf{0}$ ,the 鈥渙ld fashioned way:

$\mathbf{鉄�}\mathbf{H}\mathbf{鉄�}\mathbf{=}鈭�唯{(x,0)}^{鈭�}\widehat{H}唯(x,0)\mathrm{dx}\mathbf{.}$

Compare the result obtained in Example 2.3. Note: Because is independent of time, there is no loss of generality in using$\mathbf{t}\mathbf{=}\mathbf{0}$