58. What is the mathematical operator that represents time evolution in quantum mechanics? A. Unitary operator B. Exponential function C. Integral operator D. Differential operator
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If two operators act on a wave function as indicated by A B f(x), it is important to carry out the operations in succession, with the first operation being that nearest to the function. Mathematically, AB f(x) = A (B f(x)) and A^2 f(x) = A (A f(x)). Evaluate the following successive operations AB f(x). The operators A and B are listed in the first two columns and f(x) is listed in the third column. a. d/dx x^2 e^{-ikx} b. x^2 d/dx e^{-ikx} c. d^2/dΘ^2 d/dΘ 3cos^2Θ - 1 d. d/dΘ d^2/dΘ^2 3cos^2Θ - 1 Are your answers to parts (a) and (b) identical? Are your answers to parts (c) and (d) identical? As we will learn in Chapter 6, the fact that switching the order of the operators x and d/dx changes the outcome of the operation AB f(x) is the basis for the Heisenberg uncertainty principle.
Sri K.
(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 $$ langle Q angle_{t+t_{0}}=leftlanglePsi(x, t)left|e^{i hat{H} t_{0} / hbar} hat{Q}left(hat{x}, hat{p}, t+t_{0} ight) e^{-i hat{H} t_{0} / hbar} ight| Psi(x, t) ight angle $$ Use this to recover Equation 3.71. Hint: Let $t_{0}=d t$, and expand to first order in $d t$.
Which of the following operators is Hermitian? (a) (complex conjugate) (b) (c) (d) (e) (in two dimensions) (f) (in one dimension) (g) (in spherical polar coordinates) (h)
Shaiju T.
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