91Ó°ÊÓ

Show that if $\mathbf{⟨}\mathbf{h}\mathbf{âˆ\pounds }\widehat{\mathbf{Q}}\mathbf{h}\mathbf{âŸ\copyright }\mathbf{=}\mathbf{⟨}\widehat{\mathbf{Q}}\mathbf{h}\mathbf{âˆ\pounds }\mathbf{h}\mathbf{âŸ\copyright }$for all functions$\mathbf{h}$(in Hilbert space), then$\mathbf{⟨}\mathbf{f}\mathbf{âˆ\pounds }\widehat{\mathbf{Q}}\mathbf{g}\mathbf{âŸ\copyright }\mathbf{=}\mathbf{⟨}\widehat{\mathbf{Q}}\mathbf{f}\mathbf{âˆ\pounds }\mathbf{g}\mathbf{âŸ\copyright }$for allrole="math" localid="1655395250670" $\mathit{f}$and$\mathit{g}$(i.e., the two definitions of "Hermitian" -Equations 3.16 and 3.17- are equivalent).

(a) Write down the time-dependent "Schrödinger equation" in momentum space, for a free particle, and solve it. Answer: $\mathbf{exp}\left(-i{p}^{2}t/2m\sout{h}\right)\mathbf{Φ}\left(p,0\right)\mathbf{.}$

(b) Find role="math" localid="1656051039815" $\mathbf{Φ}\mathbf{(}\mathbf{p}\mathbf{,}\mathbf{0}\mathbf{)}$for the traveling gaussian wave packet (Problem 2.43), and construct $\mathbf{Φ}\mathbf{(}\mathbf{p}\mathbf{,}\mathbf{t}\mathbf{)}$for this case. Also construct ${\left|\mathrm{Φ}(p,t)\right|}^{\mathbf{2}}$, and note that it is independent of time.

(c) Calculaterole="math" localid="1656051188971" androle="math" localid="1656051181044" by evaluating the appropriate integrals involving$\mathbf{Φ}$, and compare your answers to Problem 2.43.

(d) Show thatrole="math" localid="1656051421703" $<H>\mathbf{=}{<p>}^{\mathbf{2}}\mathbf{/}\mathbf{2}\mathbf{m}\mathbf{+}{<H>}_{\mathbf{0}}$(where the subscript denotes the stationary gaussian), and comment on this result.

(a) Show that the sum of two hermitian operators is hermitian.

(b) Suppose$\hat{\mathit{Q}}$is hermitian, and$\mathit{Î\pm }$is a complex number. Under what condition (on$\mathit{Î\pm }$) islocalid="1655970881952" $\mathit{Î\pm }\hat{\mathit{Q}}$hermitian?

(c) When is the product of two hermitian operators hermitian?

(d) Show that the position operator $\left(\hat{x}=x\right)$and the hamiltonian operator

localid="1655971048829" $\hat{\mathit{H}}\mathbf{=}\mathbf{-}\frac{{\sout{\mathit{h}}}^{\mathbf{2}}}{\mathbf{2}\mathit{m}}\frac{{\mathit{d}}^{\mathbf{2}}}{\mathit{d}{\mathit{x}}^{\mathbf{2}}}\mathbf{+}\mathit{V}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$are hermitian.

The Hermitian conjugate (or adjoint) of an operator $\widehat{\mathbf{Q}}$is the operator${\widehat{\mathbf{Q}}}^{\mathbf{†}}$such that

$\mathbf{⟨}\mathbf{f}\mathbf{âˆ\pounds }\widehat{\mathbf{Q}}\mathbf{g}\mathbf{âŸ\copyright }\mathbf{=}\mathbf{⟨}{\widehat{\mathbf{Q}}}^{\mathbf{†}}\mathbf{f}\mathbf{âˆ\pounds }\mathbf{g}\mathbf{âŸ\copyright }\mathbf{\text{ (´Ú´Ç°ù²¹±ô±ô}}\mathbf{f}\mathbf{\text{and}}\mathbf{g}\mathbf{\text{).}}$

(A Hermitian operator, then, is equal to its Hermitian conjugate:$\widehat{\mathbf{Q}}\mathbf{=}{\widehat{\mathbf{Q}}}^{\mathbf{†}}$)

(a)Find the Hermitian conjugates of x, i, and$\mathbf{d}\mathbf{/}\mathbf{dx}$.

(b) Construct the Hermitian conjugate of the harmonic oscillator raising operator,${\mathbf{a}}_{\mathbf{+}}$(Equation 2.47).

(c) Show that${\mathbf{\left(}\widehat{\mathbf{Q}}\widehat{\mathbf{R}}\mathbf{\right)}}^{\mathbf{†}}\mathbf{=}{\widehat{\mathbf{R}}}^{\mathbf{†}}{\widehat{\mathbf{Q}}}^{\mathbf{†}}$.

(a) Suppose that $f\left(x\right)$and $g\left(x\right)$are two eigenfunctions of an operator$\hat{Q}$ , with the same eigenvalue q . Show that any linear combination of f andgis itself an eigenfunction of $\hat{Q}$, with eigenvalue q .

(b) Check that $f\left(x\right)=exp\left(x\right)$and$g\left(x\right)=exp(-x)$ are eigenfunctions of the operator$d^2/d{x}^2$ , with the same eigenvalue. Construct two linear combinations of and that are orthogonal eigenfunctions on the interval$(-1.1)$ .