Question

(a) For a function $f(x)$ that can be expanded in a Taylor series, show that $$ fleft(x+x_{0} ight)=e^{i hat{p} x_{0} / hbar} f(x) $$ (where $x_{0}$ is any constant distance). For this reason, $hat{p} / hbar$ is called the generator of translations in space. Note: The exponential of an operator is defined by the power series expansion: $e^{hat{Q}} equiv 1+hat{Q}+(1 / 2) hat{Q}^{2}+(1 / 3 !) hat{Q}^{3}+ldots$ (b) If $Psi(x, t)$ satisfies the (time-dependent) Schrödinger equation, show that $$ Psileft(x, t+t_{0} ight)=e^{-i hat{H} t_{0} / hbar} Psi(x, t) $$ (where $t_{0}$ is any constant time); $-hat{H} / hbar$ is called the generator of translations in time. (c) Show that the expectation value of a dynamical variable $Q(x, p, t)$, at time $t+t_{0}$, can be written $^{34}$ $$ langle Q angle_{t+t_{0}}=leftlanglePsi(x, t)left|e^{i hat{H} t_{0} / hbar} hat{Q}left(hat{x}, hat{p}, t+t_{0} ight) e^{-i hat{H} t_{n} / hbar} ight| Psi(x, t) ight angle $$ Use this to recover Equation 3.71. Hint: Let $t_{0}=d t$, and expand to first order in $d t$.

          (a) For a function $f(x)$ that can be expanded in a Taylor series, show that
$$
fleft(x+x_{0}
ight)=e^{i hat{p} x_{0} / hbar} f(x)
$$
(where $x_{0}$ is any constant distance). For this reason, $hat{p} / hbar$ is called the generator of translations in space. Note: The exponential of an operator is defined by the power series expansion: $e^{hat{Q}} equiv 1+hat{Q}+(1 / 2) hat{Q}^{2}+(1 / 3 !) hat{Q}^{3}+ldots$
(b) If $Psi(x, t)$ satisfies the (time-dependent) Schrödinger equation, show that
$$
Psileft(x, t+t_{0}
ight)=e^{-i hat{H} t_{0} / hbar} Psi(x, t)
$$
(where $t_{0}$ is any constant time); $-hat{H} / hbar$ is called the generator of translations in time. (c) Show that the expectation value of a dynamical variable $Q(x, p, t)$, at time $t+t_{0}$, can be written $^{34}$
$$
langle Q
angle_{t+t_{0}}=leftlanglePsi(x, t)left|e^{i hat{H} t_{0} / hbar} hat{Q}left(hat{x}, hat{p}, t+t_{0}
ight) e^{-i hat{H} t_{n} / hbar}
ight| Psi(x, t)
ight
angle
$$
Use this to recover Equation 3.71. Hint: Let $t_{0}=d t$, and expand to first order in $d t$.

Added by Josep T.

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