91Ó°ÊÓ

Find the momentum-space wave function, $\mathit{Φ}\mathbf{(}\mathit{p}\mathbf{,}\mathbf{}\mathit{t}\mathbf{)}$,for a particle in the ground state of the harmonic oscillator. What is the probability (to 2significant digits) that a measurement of p on a particle in this state would yield a value outside the classical range (for the same energy)? Hint: Look in a math table under "Normal Distribution" or "Error Function" for the numerical part-or use Mathematica.

Consider the operator $\hat{\mathbf{Q}}\mathbf{=}{\mathit{d}}^{\mathbf{2}}\mathbf{/}\mathit{d}{\mathit{Ï•}}^{\mathbf{2}}$, where (as in Example 3.1)$\mathit{Ï•}$ is the azimuthal angle in polar coordinates, and the functions are subject to Equation 3.26. Is $\hat{\mathbf{Q}}$Hermitian? Find its eigenfunctions and eigenvalues. What is the spectrum of $\hat{\mathbf{Q}}$? Is the spectrum degenerate?

(a) Check that the eigenvalues of the hermitian operator in Example 3.1 are real. Show that the eigenfunctions (for distinct eigenvalues) are orthogonal.

(b) Do the same for the operator in Problem 3.6.

(a) Show that the set of all square-integrable functions is a vector space (refer to Section A.1 for the definition). Hint: The main problem is to show that the sum of two square-integrable functions is itself square-integrable. Use Equation 3.7. Is the set of all normalized functions a vector space?

(b) Show that the integral in Equation 3.6satisfies the conditions for an inner product (Section A.2).

Show that projection operators are idempotent: ${\hat{\mathbf{P}}}^{\mathbf{2}}\mathit{=}\hat{\mathbf{P}}$. Determine the eigenvalues of $\hat{\mathbf{P}}$ , and characterize its eigenvectors.

Consider a three-dimensional vector space spanned by an Orthonormal basis $\overline{)\mathbf{1}\mathbf{>}\mathbf{,}\overline{)\mathbf{2}\mathbf{>}\mathbf{,}\overline{)\mathbf{3}\mathbf{>}}}}$. Kets $\overline{)\mathbf{Î\pm }\mathbf{>}}$and $\overline{)\mathit{β}\mathbf{>}}$are given by

$\mathbf{|}\mathbf{}\mathit{Î\pm }\mathbf{âŸ\copyright }\mathbf{=}\mathit{i}\mathbf{|}\mathbf{}\mathbf{1}\mathbf{âŸ\copyright }\mathbf{-}\mathbf{2}\mathbf{|}\mathbf{}\mathbf{2}\mathbf{âŸ\copyright }\mathbf{-}\mathit{i}\mathbf{|}\mathbf{}\mathbf{3}\mathbf{âŸ\copyright }\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{|}\mathit{β}\mathbf{>}\mathbf{=}\mathit{i}\mathbf{|}\mathbf{}\mathbf{1}\mathbf{âŸ\copyright }\mathbf{+}\mathbf{2}\mathbf{|}\mathbf{}\mathbf{3}\mathbf{âŸ\copyright }\mathbf{.}$

(a)Construct$\mathbf{<}\overline{)\mathbf{Î\pm }}$and $\mathbf{<}\overline{)\mathbf{β}}$(in terms of the dual basis

$\mathbf{}\mathbf{⟨}\mathbf{1}\mathbf{|}\mathbf{,}\mathbf{⟨}\mathbf{2}\mathbf{|}\mathbf{,}\mathbf{⟨}\mathbf{3}\mathbf{\left|}\mathbf{\right)}\mathbf{.}$
(b) Find $\mathbf{⟨}\mathit{Î\pm }\mathbf{âˆ\pounds }\mathbf{}\mathit{β}\mathbf{âŸ\copyright }$and$\mathbf{}\mathbf{⟨}\mathit{β}\mathbf{âˆ\pounds }\mathbf{}\mathit{Î\pm }\mathbf{âŸ\copyright }\mathbf{,}$and confirm that

$\mathbf{⟨}\mathit{β}\mathbf{âˆ\pounds }\mathit{Î\pm }\mathbf{âŸ\copyright }\mathbf{=}\mathbf{⟨}\mathit{Î\pm }\mathbf{âˆ\pounds }\mathit{β}{\mathbf{âŸ\copyright }}^{\mathbf{*}}\mathbf{.}$
(c)Find all nine matrix elements of the operator$\stackrel{\mathbf{Áåœ}}{\mathbf{A}}\mathbf{}\mathbf{≡}\mathbf{|}\mathbf{}\mathit{Î\pm }\mathbf{âŸ\copyright }\mathbf{⟨}\mathit{β}\mathbf{|}$, in this basis, and construct the matrix A. Is it hermitian?

Let $\hat{\mathbf{Q}}$be an operator with a complete set of orthonormal eigenvectors:localid="1658131083682" $\hat{\mathbf{Q}}{\overline{)\mathbf{e}}}_{\mathbf{n}}\mathbf{>}\mathbf{=}{\mathit{q}}_{\mathbf{n}}\overline{){\mathbf{e}}_{\mathbf{n}}}$(n=1,2,3,....) Show that$\hat{\mathbf{Q}}$can be written in terms of its spectral decomposition:$\hat{\mathbf{Q}}\mathbf{=}\underset{\mathbf{n}}{\mathbf{∑}}{\mathit{q}}_{\mathbf{n}}\overline{){\mathbf{e}}_{\mathbf{n}}\mathbf{>}\mathbf{<}{\mathbf{e}}_{\mathbf{n}}\mathbf{|}}$

Hint: An operator is characterized by its action on all possible vectors, so what you must show is that$\hat{\mathbf{Q}}\mathbf{=}\left\{\underset{n}{∑}{q}_n\overline{)e_n><e_n|}\right\}$ for any vector $\overline{)\mathbf{Î\pm }\mathbf{>}}$.

Legendre polynomials. Use the Gram Schmidt procedure (ProblemA.4) to orthonormalize the functions $\mathbf{1}\mathbf{,}\mathit{x}\mathbf{,}{\mathit{x}}^{\mathbf{2}}\mathbf{,}\mathbf{and}\mathbf{}{\mathbf{x}}^{\mathbf{3}}$, on the interval$\mathbf{-}\mathbf{1}\mathbf{≤}\mathit{x}\mathbf{≤}\mathbf{1}$. You may recognize the results-they are (apart from the normalization)${}^{\mathbf{30}}$Legendre polynomials (Table 4.1 )

Suppose$\mathrm{Ψ}(x,0)=\frac{A}{x^2+a^2}.(-∞<x<∞)$for constants Aand a.

(a) Determine A, by normalizing$\mathrm{Ψ}(x,0)$.

(b) Find, and${\mathrm{σ}}_x$(at time$t=0$).

(c) Find the momentum space wave function$\mathrm{Φ}(p,0)$, and check that it is normalized.

(d) Use$\mathrm{Φ}(p,0)$to calculate, and${\mathrm{σ}}_p$(at time$t=0$).

(e) Check the Heisenberg uncertainty principle for this state.

Coherent states of the harmonic oscillator. Among the stationary states of the harmonic oscillator (Equation 2.67) only n = 0 hits the uncertainty limit $\left({\mathrm{σ}}_x{\mathrm{σ}}_p=\sout{h}/2\right)$; in general, ${\mathbf{σ}}_{\mathbf{x}}{\mathbf{σ}}_{\mathbf{p}}\mathbf{=}\left(2n+1\right)\sout{\mathbf{h}}\mathbf{}\mathbf{/}\mathbf{2}$, as you found in Problem 2.12. But certain linear combinations (known as coherent states) also minimize the uncertainty product. They are (as it turns out) Eigen functions of the lowering operator

${\mathit{ψ}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{n}\mathbf{!}}}{\left({\hat{a}}_{+}\right)}^{\mathbf{n}}{\mathit{ψ}}_{\mathbf{0}}$(2.68).

$\mathit{a}\mathit{\_}\left|\mathrm{Î\pm }>=\mathrm{Î\pm }\right|\mathit{a}\mathbf{>}$(the Eigen value α can be any complex number).

(a)Calculate $<x>\mathbf{,}<x^2>\mathbf{,}<p>\mathbf{,}<p^2>$in the state |α〉. Hint: Use the technique in Example 2.5, and remember that is the Hermitian conjugate of $\mathit{a}\mathbf{-}$. Do not assume α is real.

(b) Find ${\mathit{σ}}_{\mathbf{x}}$; show that ${\mathit{σ}}_{\mathbf{x}}{\mathit{σ}}_{\mathbf{p}}\mathbf{=}\sout{\mathbf{h}}\mathbf{}\mathbf{/}\mathbf{2}$.

(c) Like any other wave function, a coherent state can be expanded in terms of energy Eigen states: $\left|\mathrm{Î\pm }>=\underset{n=0}{\overset{∞}{∑}}{C}_n\right|\mathbf{}\mathit{n}\mathbf{>}\mathbf{.}$

Show that the expansion coefficients are${\mathit{c}}_{\mathbf{n}}\mathbf{=}\frac{{\mathbf{Î\pm }}^{\mathbf{n}}}{\sqrt{\mathbf{n}\mathbf{!}}}{\mathit{c}}_{\mathbf{0}}\mathbf{.}$

(d) Determine by normalizing |α〉. Answer: exp$\left(-{\left|\mathrm{Î\pm }\right|}^2/2\right)$

(e) Now put in the time dependence: $\left|n>→e^{-i{E}_{n}tI\sout{h}}\right|\mathbf{}\mathit{n}\mathbf{>}$,

and show that $\left|\mathrm{Î\pm }\left(t\right)\right|$remains an Eigen state of $\mathit{a}\mathbf{-}$, but the Eigen value evolves in time:$\mathit{Î\pm }\left(t\right)\mathbf{=}{\mathit{e}}^{\mathbf{-}\mathbf{i}\mathbf{Ó\neg }\mathbf{t}}$ So a coherent state stays coherent, and continues to minimize the uncertainty product.

(f) Is the ground state $\left(\left|n=0>\right|\right)$itself a coherent state? If so, what is the Eigen value?