91影视

Question: Find the probability current, $\mathbf{J}$ (Problem 1.14) for the free particle wave function Equation $\mathbf{2}\mathbf{.}\mathbf{94}$. Which direction does the probability flow?

Prove the following three theorem;

a) For normalizable solutions the separation constant E must be real as ${\mathit{E}}_{\mathbf{0}}\mathbf{+}\mathit{i}\mathit{蟿}$and show that if equation 1.20 is to hold for all $\mathit{t}\mathbf{,}\mathit{蟿}$ must be zero.

b) The time - independent wave function localid="1658117146660" $\mathit{蠄}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ can always be taken to be real, This doesn鈥檛 mean that every solution to the time-independent Schrodinger equation is real; what it says is that if you鈥檝e got one that is not, it can always be expressed as a linear combination of solutions that are . So, you might as well stick to$\mathit{蠄}$ 鈥檚 that are real

c) If is an even function then $\mathit{蠄}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$can always be taken to be either even or odd

This problem is designed to guide you through a 鈥減roof鈥� of Plancherel鈥檚 theorem, by starting with the theory of ordinary Fourier series on a finite interval, and allowing that interval to expand to infinity.

(a) Dirichlet鈥檚 theorem says that 鈥渁ny鈥� function f(x) on the interval $\left[-a,+a\right]$can be expanded as a Fourier series:

$\mathbf{f}\left(x\right)\mathbf{=}\underset{\mathbf{n}\mathbf{=}\mathbf{0}}{\overset{\mathbf{鈭�}}{\mathbf{鈭�}}}\left[a_n\sin \left(\frac{\mathrm{苍蟺虫}}{a}\right)+b_n\cos \left(\frac{\mathrm{苍蟺虫}}{a}\right)\right]$

Show that this can be written equivalently as

$\mathbf{f}\left(x\right)\mathbf{=}\underset{\mathbf{n}\mathbf{=}\mathbf{-}\mathbf{鈭�}}{\overset{\mathbf{鈭�}}{\mathbf{鈭�}}}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{i苍蟺虫}\mathbf{/}\mathbf{a}}$.

What is ${\mathbf{c}}_{\mathbf{n}}$, in terms of ${\mathbf{a}}_{\mathbf{n}}$and ${\mathbf{b}}_{\mathbf{n}}$?

(b) Show (by appropriate modification of Fourier鈥檚 trick) that

${\mathbf{c}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{a}}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{a}}^{\mathbf{+}\mathbf{a}}\mathbf{f}\left(x\right){\mathbf{e}}^{\mathbf{-}\mathbf{i苍蟺虫}\mathbf{/}\mathbf{a}}\mathbf{dx}$

(c) Eliminate n and ${\mathbf{c}}_{\mathbf{n}}$in favor of the new variables $\mathbf{k}\mathbf{=}\left(\mathrm{苍蟿蟿}/a\right)\mathbf{}\mathbf{andF}\mathbf{}\left(k\right)\mathbf{=}\sqrt{\mathbf{2}\mathbf{/}\mathbf{蟺}}{\mathbf{ac}}_{\mathbf{n}}$. Show that (a) and (b) now become

$\mathbf{f}\left(x\right)\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}\mathbf{蟺}}}\underset{\mathbf{n}\mathbf{=}\mathbf{-}\mathbf{鈭�}}{\overset{\mathbf{鈭�}}{\mathbf{鈭�}}}\mathbf{F}\left(k\right){\mathbf{e}}^{\mathbf{ikx}}\mathbf{鈭�}\mathbf{k}\mathbf{;}\mathbf{}\mathbf{F}\left(k\right)\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}\mathbf{蟺}}}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{a}}^{\mathbf{+}\mathbf{a}}\mathbf{f}\left(x\right)\mathbf{.}{\mathbf{e}}^{\mathbf{ikx}}\mathbf{dx}$.

where $\mathbf{鈭�}\mathbf{k}$is the increment in k from one n to the next.

(d) Take the limit $\mathbf{a}\mathbf{鈫�}\mathbf{鈭�}$to obtain Plancherel鈥檚 theorem. Comment: In view of their quite different origins, it is surprising (and delightful) that the two formulas鈥攐ne for F(k) in terms of f(x), the other for f(x) terms of F(k) 鈥攈ave such a similar structure in the limit $\mathbf{a}\mathbf{鈫�}\mathbf{鈭�}$.

A free particle has the initial wave function
$\mathbf{蠄}\left(x,0\right)\mathbf{=}{\mathbf{Ae}}^{\mathbf{-}\mathbf{a}\left|x\right|}\mathbf{,}$

where A and a are positive real constants.

(a)Normalize$\mathbf{蠄}\left(x,0\right)\mathbf{.}$

(b) Find$\mathbf{蠒}\left(k\right)$.

(c) Construct $\mathbf{蠄}\left(x,t\right)$,in the form of an integral.

(d) Discuss the limiting cases very large, and a very small.

The gaussian wave packet. A free particle has the initial wave function

$\mathbf{Y}\mathbf{(}\mathit{x}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{=}\mathit{A}{\mathit{e}}^{\mathbf{-}\mathbf{a}{\mathbf{x}}^{\mathbf{2}}}$

where$\mathit{A}$and are constants ( is real and positive).

(a) Normalize$\mathit{Y}\mathbf{(}\mathit{x}\mathbf{,}\mathbf{0}\mathbf{)}$

(b) Find $\mathbf{Y}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$. Hint: Integrals of the form

${\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{+}\mathbf{鈭�}}{\mathit{e}}^{\mathbf{-}\left(a{x}^2+bx\right)}\mathit{d}\mathit{x}$

Can be handled by 鈥渃ompleting the square鈥�: Let$\mathit{y}\mathbf{=}\sqrt{\mathbf{a}}\left[x+\left(bl2a\right)\right]$, and note that$\left(a{x}^2+bx\right)\mathbf{=}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\left(b^{2}l4a\right)$. Answer:

localid="1658297483210" $\mathit{Y}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}\mathbf{=}{\mathbf{\left(}\frac{\mathbf{2}\mathbf{a}}{\mathbf{蟺}}\mathbf{\right)}}^{\mathbf{1}\mathbf{/}\mathbf{4}}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{e}{\mathbf{x}}^{\mathbf{2}}\mathbf{l}\mathbf{\left[}\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{i}\mathbf{h}\mathbf{a}\mathbf{t}\mathbf{}\mathbf{l}\mathbf{m}\mathbf{)}\mathbf{\right]}}}{\sqrt{\mathbf{1}\mathbf{+}\mathbf{(}\mathbf{2}\mathbf{i}\sout{\mathbf{h}}\mathbf{a}\mathbf{t}\mathbf{}\mathbf{l}\mathbf{}\mathbf{m}\mathbf{)}}}$

(c) Find . Express your answer in terms of the quantity

localid="1658297497509" $\mathbf{蝇}\mathbf{=}\sqrt{\frac{\mathbf{a}}{\mathbf{1}\mathbf{+}{\mathbf{(}\mathbf{2}\mathbf{i}\mathbf{}\sout{\mathbf{h}}\mathbf{at}\mathbf{}\mathbf{l}\mathbf{}\mathbf{m}\mathbf{)}}^{\mathbf{2}}}}$

Sketchlocalid="1658124147567" ${\left|Y\right|}^{\mathbf{2}}$(as a function of $\mathit{x}$) at $\mathbf{t}\mathbf{=}\mathbf{0}$, and again for some very large $\mathbf{t}$. Qualitatively, what happens to ${\left|Y\right|}^{\mathbf{2}}$, as time goes on?

(d) Find $<x>\mathbf{,}<p>\mathbf{,}<x^2>\mathbf{,}<p^2>\mathbf{,}{\mathit{蟽}}_{\mathbf{x}}$and ${\mathit{蟽}}_{\mathbf{P}}$. Partial answer:localid="1658297458579" $\mathbf{<}{\mathbf{p}}^{\mathbf{2}}\mathbf{>}\mathbf{=}\mathit{a}{\sout{\mathbf{h}}}^{\mathbf{2}}$, but it may take some algebra to reduce it to this simple form.

(e) Does the uncertainty principle hold? At what time $\mathbf{t}$does the system come

closest to the uncertainty limit?

Evaluate the following integrals:

(a)${\mathbf{鈭�}}_{\mathbf{-}\mathbf{3}}^{\mathbf{+}\mathbf{1}}\mathbf{(}{\mathit{x}}^{\mathbf{3}}\mathbf{-}\mathbf{3}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{-}\mathbf{1}\mathbf{)}\mathit{未}\mathbf{(}\mathit{x}\mathbf{+}\mathbf{2}\mathbf{)}\mathit{d}\mathit{x}$.

(b).${\mathbf{鈭�}}_{\mathbf{0}}^{\mathbf{鈭�}}\mathbf{[}\mathbf{cos}\left(3x\right)\mathbf{+}\mathbf{2}\mathbf{]}\mathbf{未}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{蟺}\mathbf{)}\mathbf{dx}$

(c)${\mathbf{鈭�}}_{\mathbf{\_}\mathbf{1}}^{\mathbf{+}\mathbf{1}}\mathbf{exp}\mathbf{(}\mathbf{lxl}\mathbf{+}\mathbf{3}\mathbf{)}\mathit{未}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{2}\mathbf{)}\mathit{d}\mathit{x}$

Delta functions live under integral signs, and two expressions $\left(D_1\left(x\right)and{D}_2\left(x\right)\right)$involving delta functions are said to be equal if

${\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{+}\mathbf{鈭�}}\mathit{f}\left(x\right){\mathit{D}}_{\mathit{1}}\left(x\right)\mathit{d}\mathit{x}\mathbf{=}{\mathbf{鈭�}}_{\mathbf{-}\mathbf{鈭�}}^{\mathbf{+}\mathbf{鈭�}}\mathit{f}\left(x\right){\mathit{D}}_{\mathit{2}}\left(x\right)\mathit{d}\mathit{x}$for every (ordinary) function f(x).

(a) Show that

$\mathit{未}\left(cx\right)\mathbf{=}\frac{\mathbf{1}}{\left|c\right|}\mathit{未}\left(x\right)$(2.145)

where c is a real constant. (Be sure to check the case where c is negative.)

(b) Let $\mathit{胃}\left(x\right)$ be the step function:

$\mathit{胃}\left(x\right)\mathbf{鈮�}\{\begin{array}{l}1,x>0\\ 0,x>0\end{array}$(2.146).

(In the rare case where it actually matters, we define $\mathit{胃}\left(0\right)$ to be 1/2.) Show that $\mathit{d}\mathit{胃}\mathit{}\mathit{l}\mathit{}\mathit{d}\mathit{x}\mathit{}\mathit{=}\mathit{}\mathit{未}$

Check the uncertainty principle for the wave function in the equation? Equation 2.129.

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 ${\mathbf{蠄}}_2$ 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

What is the Fourier transform $\mathbf{未}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ ? Using Plancherel鈥檚 theorem shows that$\mathbf{未}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}\mathbf{蟺}}\underset{鈭�\mathrm{鈭�}}{\overset{\mathrm{鈭�}}{鈭�}}{e}^{\mathrm{ikx}}\mathbf{dk}$.