answers-03-29-2011

answers-03-29-2011 - Answers to Physics 176 One-Minute...

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Answers to Physics 176 One-Minute Questionnaires March 29, 2011 Can you have a probability density that is a function of vectors instead of the components of a vector? The answer is yes because there is no practical difference between defining a probability density (or any function) in terms of the components of a vector versus the vector itself. For example, a molecule can be described by two 3-vectors x = ( x 1 , x 2 , x 3 ) and v = ( v 1 , v 2 , v 3 ) or, completely equivalently, by the six vector components ( x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ), so we could write the probability density as a function of vectors like this, D ( x , v ), or write it equivalently as D ( x 1 , x 2 , x 3 , v 1 , v 2 , v 3 ). A way to see that these are equivalent is that vector components like x i and v i can be obtained by operations carried out on the vectors x and v themselves, for example x 1 = x · ˆ x, v 2 = v · ˆ y, (1) where ˆ x is a unit vector along the positive x -axis and ˆ y is a unit vector along the positive y -axis. So any expression involving the components of the vectors can be alternatively interpreted as some operation involving the vectors themselves and vice versa. Are continuous energies (like (1 / 2) PV 2 , mgh ), actually continuous or are they discretized in units of Planck’s constant? That’s a tricky question. If the universe is finite in size (it is not known one way or the other but most scientists believe it is), then all energy levels must be quantized because of the properties of the Schrodinger equation in a finite region of space. But if you calculate the order of magnitude of the energy difference Δ E between two energy levels of an electron (the lightest mass stable particle produces the largest energy difference) in a box of size L (see Eq. (A.14) on page 369 of Schroeder) where L is of order 10 billion light years, one gets Δ E h 2 8 mL 2 (2) ( 7 × 10 - 34 J · s ) 2 8 × (9 × 10 - 31 kg) × ((10 × 10 9 ly) × 10 16 m / ly) 2 (3) 10 - 71 eV , (4) 1
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which is far far tinier than any experimental device can detect. Even for a one centimeter cube of some metal like copper, the energy spacings between the electron energy levels are too small to be measureable.
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