91影视

A particle is in the ground state of the harmonic oscillator with classical frequency $蝇$, when suddenly the spring constant quadruples, so $蝇\text{'}=2蝇$, without initially changing the wave function (of course, $唯$will now evolve differently, because the Hamiltonian has changed). What is the probability that a measurement of the energy would still return the value $\frac{\mathrm{鈩�}蝇}{2}$? What is the probability of getting $\mathrm{鈩�}蝇$?

Find , for the nth stationary state of the harmonic oscillator, using the method of Example $2.5$. Check that the uncertainty principle is satisfied.

A particle in the harmonic oscillator potential starts out in the state$\mathit{唯}\mathbf{(}\mathit{x}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{=}\mathit{A}\mathbf{\left[}\mathbf{3}{\mathbf{蠄}}_{\mathbf{0}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}\mathbf{4}{\mathbf{蠄}}_{\mathbf{1}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{\right]}$

a) Find $\mathit{A}$.

b) Construct $\mathit{唯}\mathbf{(}\mathit{x}\mathbf{,}\mathit{t}\mathbf{)}$and$\left|唯(x,t{)}^2\right|$

c) Find $\mathbf{鉄�}\mathit{x}\mathbf{鉄�}$and $\mathbf{鉄�}\mathit{p}\mathbf{鉄�}$. Don't get too excited if they oscillate at the classical frequency; what would it have been had I specified ${\mathit{蠄}}_{\mathbf{2}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$, instead of ${\mathit{蠄}}_{\mathbf{1}}\mathbf{\left(}\mathit{x}\mathbf{\right)}$?Check that Ehrenfest's theorem holds for this wave function.

d) If you measured the energy of this particle, what values might you get, and with what probabilities?

Show that $\left[A{e}^{ikx}+B{e}^{-ikx}\right]$ and $\mathbf{\left[}\mathit{C}\mathit{c}\mathit{o}\mathit{s}\mathbf{\right(}\mathit{k}\mathit{x}\mathbf{)}\mathbf{+}\mathit{D}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{k}\mathit{x}\mathbf{\left)}\mathbf{\right]}$ are equivalent ways of writing the same function of $\mathit{x}$, and determine the constants $\mathit{C}$ and $\mathit{D}$ in terms of $\mathit{A}$and $\mathit{B}$, and vice versa.

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

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?

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

Consider the double delta-function potential$\mathit{V}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{-}\mathit{伪}\left[未\left(x+a\right)+未\left(x-a\right)\right]$Whereand are positive constants

(a) Sketch this potential.

(b) How many bound states does it possess? Find the allowed energies, for$\mathit{伪}\mathbf{=}\mathit{魔}\mathbf{/}\mathit{m}\mathit{a}$and for$\mathit{伪}\mathbf{=}{\mathit{魔}}^{\mathbf{2}}\mathbf{/}\mathbf{4}\mathit{m}\mathit{a}$, and sketch the wave functions.