91影视

For the distribution of ages in the example in Section 1.3.1:

(a) Compute$\mathbf{鉄�}{\mathbf{j}}^{\mathbf{2}}\mathbf{鉄�}$ and${\mathbf{鉄�}\mathbf{j}\mathbf{鉄�}}^{\mathbf{2}}$ .

(b) Determine 鈭�j for each j, and use Equation 1.11 to compute the standard deviation.

(c) Use your results in (a) and (b) to check Equation 1.12.

(a) Find the standard deviation of the distribution in Example 1.1.

(b) What is the probability that a photograph, selected at random, would show a distance x more than one standard deviation away from the average?

Consider the Gaussian distribution

$\mathbf{\text{蚁}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{Ae}}^{\mathbf{鈭�}\mathbf{位}{\mathbf{(}\mathbf{x}\mathbf{鈭�}\mathbf{a}\mathbf{)}}^{\mathbf{2}}}$

where A, a, and 位 are positive real constants. (Look up any integrals you need.)

(a) Use Equation 1.16 to determine A.

(b) Find銆坸銆�,銆坸²銆�,and 蟽.

(c) Sketch the graph of 蚁(x).

At time t = 0 a particle is represented by the wave function

$\mathbf{蠄}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{)}\mathbf{=}\{\begin{array}{ll}A(x,0),& 0鈮�x鈮�a,\\ A(b鈭�x)/(b鈭�a),& a鈮�x鈮�b,\\ 0,& \mathrm{otherwise},\end{array}$where A, a, and b are (positive) constants.

(a) Normalize $\mathbf{蠄}$(that is, find A, in terms of a and b).

(b) Sketch $\mathbf{蠄}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{0}\mathbf{)}$, as a function of x.

(c) Where is the particle most likely to be found, at t = 0?.

(d) What is the probability of finding the particle to the left of a? Check your result in the limiting cases b = a and b= 2a.

(e) What is the expectation value of x?

Consider the wave function
${蠄}_{(x,t)}=A{e}^{-位\left|x\right|}{e}^{-i蝇t}$

whereA, 位, and 蝇 are positive real constants. (We鈥檒l see in Chapter for what potential (V) this wave function satisfies the Schr枚dingerequation.)

(a) Normalize$蠄$ .

(b) Determine the expectation values of$x$ and $x^2$.

(c) Find the standard deviation of . Sketch the graph of${\left|\mathbf{唯}\right|}^{\mathbf{2}}$ , as a function ofx, and mark the points $(鉄�x鉄�+蟽)$and $(鉄�x鉄�-蟽)$, to illustrate the sense in which蟽 represents the 鈥渟pread鈥� inx. What is the probability that the particle would be found outside this range?

Why can鈥檛 you do integration-by-parts directly on the middle expression in Equation -1.29 pull the time derivative over onto x, note that$\mathbf{鈭�}\mathbf{x}\mathbf{/}\mathbf{鈭�}\mathbf{t}\mathbf{=}\mathbf{0}$ , and conclude that$\mathbf{d}<x>\mathbf{/}\mathbf{dt}\mathbf{=}\mathbf{0}$ ?

Calculate d銆坧銆�/dt. Answer:

This is an instance of Ehrenfest鈥檚 theorem, which asserts that expectation values obey the classical laws

Suppose you add a constant${\mathbf{V}}_{\mathbf{0}}$ to the potential energy (by 鈥渃onstant鈥� I mean independent ofxas well as t). In classical mechanics this doesn鈥檛 change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor:$\mathbf{exp}\mathbf{(}\mathbf{-}{\mathbf{iV}}_{\mathbf{0}}\mathbf{t}\mathbf{/}\sout{\mathbf{h}}\mathbf{)}$. What effect does this have on the expectation value of a dynamical variable?

A particle of mass m is in the state:

$\mathbf{蠄}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{Ae}}^{\mathbf{鈭�}\mathbf{a}\mathbf{[}\mathbf{(}{\mathbf{mx}}^{\mathbf{2}}\mathbf{/}\sout{\mathbf{h}}\mathbf{)}\mathbf{+}\mathbf{it}\mathbf{]}}$

where A and a are positive real constants.

(a) Find A.

(b) For what potential energy function, V(x), is this a solution to the Schr枚dinger equation?

(c) Calculate the expectation values of x,${\mathit{x}}^{\mathbf{2}}$ , p, and${\mathit{p}}^{\mathbf{2}}$ .

(d) Find 蟽_x and 蟽_p. Is their product consistent with the uncertainty principle?