91影视

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?