Download Principia : Vol. 1 The Motion of Bodies by Isaac Newton, Florian Cajori, Andrew Motte PDF
By Isaac Newton, Florian Cajori, Andrew Motte
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Additional info for Principia : Vol. 1 The Motion of Bodies
Example text
30 2. Light Wavess Fig. 12. Time development of the quantity E c E c∗ (which except for a factor n 2 is proportional to the average energy density) in a wave packet of light incident on a glass plate. 6 Wave Packet Traveling through a Glass Plate Let us study a wave packet that is relatively narrow in space, that is, one containing a wide range of frequencies. 12) shows that, as expected, at the front surface of the glass plate part of the packet is reflected. Another part enters the plate, where it is compressed and travels with reduced speed.
The spectral function f (k) specifies the amplitude of the harmonic wave with wave number k and circular frequency ω = ck. We now consider a particularly simple spectral function which is significantly different from zero in the neighborhood of the wave number k0 . We choose the Gaussian function (k − k0 )2 1 exp − f (k) = √ 2σk2 2π σk . It describes a bell-shaped spectral function which has its maximum value at k = k0 ; we assume the value of k0 to be positive, k0 > 0. The width of the region in which the function f (k) is different from zero is characterized by the parameter σk .
Putting the two findings together, we have discovered that the momentum expectation value of a free, unaccelerated Gaussian wave packet is the same as the momentum of a free, unaccelerated particle of mass m and velocity v0 in classical mechanics: p = p0 = mv0 . The expectation value of momentum can also be calculated directly from the wave function ψ(x, t). We have the simple relation h¯ ∂ i 1 h¯ ∂ exp − (Et − px) ψ p (x − x0 , t) = 1/2 h¯ i ∂x i ∂ x (2π h¯ ) = pψ p (x − x0 , t) . This relation translates the momentum variable p into the momentum operator p→ h¯ ∂ i ∂x .