Q20P Show that the energy-time uncert... [FREE SOLUTION]

91影视

Introduction to Quantum Mechanics

David J. Griffiths

Physics

2nd Edition 路 469 pages

ISBN: 9780131118928

Chapter 3: Q20P (page 118)

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

Short Answer

Expert verified

The energy-time uncertainty principle reduces to $蟽_{H}^{2} 蟽_{x}^{2} = \frac{魔^{2}}{4 m^{2}} {<p>}^{2}$ .

Step by step solution

The generalized uncertainty principle for two operators.

The generalized uncertainty principle for two operators is:

$未_{A}^{2} 未_{B}^{2} 鈮� {(\frac{1}{2 i} < [\overset{铃�}{A}, \overset{铃�}{B}] >)}^{2}$

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle.

From the energy-time uncertainty principle, an operator Q satisfies the equation:

$\frac{d}{d t} <Q> = \frac{i}{魔} <[H, Q]> + <\frac{鈭� Q}{鈭� t}> for the position operator, Q = x, the time derivative of the position is zero, since x and t are independent variables, \frac{d}{dt} <x> = \frac{i}{魔} <[H, x]> let \overset{铃�}{A} = \overset{铃�}{H} and \overset{铃�}{B} = \overset{铃�}{x} and use (2), to get; 蟽_{H}^{2} 蟽_{x}^{2} 鈮� {(- \frac{魔}{2} \frac{d <x>}{dt})}^{2} = \frac{魔^{2}}{4 m^{2}} {(m \frac{d <x>}{dt})}^{2} = \frac{魔^{2}}{4 m^{2}} {<p>}^{2} 蟽_{H}^{2} 蟽_{x}^{2} = \frac{魔^{2}}{4 m^{2}} {<p>}^{2} Thus, the energy - time uncertainty principle reduces to 蟽_{H}^{2} 蟽_{x}^{2} = \frac{魔^{2}}{4 m^{2}} {<p>}^{2} .$

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91影视

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle (Problem 3.14), when the observable in question is x.

Short Answer

Step by step solution

The generalized uncertainty principle for two operators.

Show that the energy-time uncertainty principle reduces to the "your name" uncertainty principle.

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