91Ó°ÊÓ

Calculate $\left(x\right)\mathbf{,}\left(x^2\right)\mathbf{,}\left(p\right)\mathbf{,}\left(p^2\right)\mathbf{,}{\mathit{σ}}_{\mathbf{x}}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}{\mathit{σ}}_{\mathbf{p}}$,for the nth stationary state of the infinite square well. Check that the uncertainty principle is satisfied. Which state comes closest to the uncertainty limit?

Consider the moving delta-function well: $\mathbf{V}\left(x,t\right)\mathbf{=}\mathbf{-}\mathbf{Î\pm δ}\left(x-\mathrm{vt}\right)$

where v is the (constant) velocity of the well. (a) Show that the time-dependent Schrödinger equation admits the exact solution $\mathbf{\text{ψ}}\left(x,t\right)\mathbf{=}\sqrt{\frac{\mathbf{³¾Î\pm }}{\sout{\mathbf{h}}}}{\mathbf{e}}^{\mathbf{-}\mathbf{³¾Î\pm }\left|x-\mathrm{vt}\right|}\mathbf{l}{\sout{\mathbf{h}}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{-}\mathbf{i}}\left[\left(E+\left(1/2\right){\mathrm{mv}}^2\right)t-\mathrm{mvx}\right]\mathbf{l}\sout{\mathbf{h}}$where $\mathbf{E}\mathbf{=}\mathbf{-}{\mathbf{³¾Î\pm }}^{\mathbf{2}}\mathbf{}\mathbf{l}\mathbf{}\mathbf{2}{\sout{\mathbf{h}}}^{\mathbf{2}}$ is the bound-state energy of the stationary delta function. Hint: Plug it in and check it! Use the result of Problem 2.24(b). (b) Find the expectation value of the Hamiltonian in this state, and comment on the result.

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 ${\mathit{ψ}}_{\mathbf{1}}\mathbf{}\mathbf{and}\mathbf{}{\mathit{ψ}}_{\mathbf{2}}$in problem 2.5:$\mathit{ψ}\left(x,0\right)\mathbf{=}\mathit{A}\left[{\mathrm{ψ}}_1\left(x\right)+e^{i\mathrm{ϕ}}{\mathrm{ψ}}_2\left(x\right)\right]$Where $\mathit{ϕ}$is some constant. Find $\mathit{ψ}\left(x,t\right)\mathbf{,}\mathbf{}{\left|\mathrm{ψ}\left(x,t\right)\right|}^{\mathbf{2}}$, and $\left(x\right)$, and compare your results with what you got before. Study the special cases $\mathit{ϕ}\mathbf{=}\raisebox{1ex}{$\mathbf{π}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.\mathbf{}\mathbf{and}\mathbf{}\mathit{ϕ}\mathbf{=}\mathbf{π}$.

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

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

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

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

(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.