Phase Velocity and Group Velocity for Beginners
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1 Phase Veloity and Group Veloity for Beginners Rodolfo A. Frino January 015 (v1) - February 015 (v3) Eletronis Engineer Degree from the National University of Mar del Plata - Argentina Copyright Rodolfo A. Frino. All rights reserved. Abstrat In this paper, first, I derive the formulas for the phase veloity and group veloity as a funtion of the total relativisti energy and the momentum of a partile. Then, I derive similar formulas as a funtion of the de Broglie and the Compton wavelengths of the partile. Finally an additional meaning of the Compton wavelength is derived from the equation of the group veloity in terms of the de Broglie and the Compton wavelengths. Keywords: phase veloity, group veloity, total relativisti energy, momentum, relativisti mass, de Broglie wavelength, Compton wavelength, Compton momentum. 1. Phase Veloity and Group Veloity as a Funtion of the Total Relativisti Energy and the Relativisti Momentum of a Partile 1.1 Phase Veloity Let us onsider the following three laws from Einstein's speial theory of relativity: a) The formula of equivalene of mass and energy b) The formula of the relativisti momentum ) The formula of the relativisti mass E m (1.1-1) p m (1.1-) where m v 1 g (1.1-3) E total relativisti energy of a partile m relativisti mass of a partile rest mass of a partile p relativisti momentum of a partile group veloity of a partile Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 1
2 speed of light in vauum Let us begin by dividing equation (1) by equation () Simplifying we get E p m m E p (1.1-4) (1.1-5) The dimensions of equation (1.1-5) tells us that the ratio s E/p is a veloity. Furthermore, beause the group veloity is always smaller than the speed of light, this veloity, s, must be greater than the speed of light,, Therefore the ratio s must be the phase veloity,. Thus, we an write E p (1.1-6) Thus we an draw the following onlusion Phase Veloity The phase veloity of any partile (massive or massless) is equal to its total relativisti energy divided by its momentum. Finally from equations (1.1-5) and (1.1-6) we get Whih an be translated into words as follows (1.1-7) The Produt of the Phase Veloity and The Group Veloity The produt between the phase veloity and the group veloity of any partile (massive or massless) equals the square of the speed of light in vauum. 1.. Group Veloity Let us onsider the Einstein's total relativisti energy formula Now we derive both sides of this equation with respet to p E p + 4 (1.-1) d dp (E ) d dp ( p + 4 ) (1.-) Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved.
3 Observing that both and are onstants we get After simple mathematis steps we get E de dp p dp dp + 0 (1.-3) de dp p E (1.-4) Substituting the denominator of the seond side, E, with the seond side of equation (1.1-1) we get de dp p m p m (1.-5) Substituting the numerator of the seond side, p, with the seond side of equation (1.1-) we get de dp m m (1.-6) Finally we swap sides to get the formula for the group veloity de dp (1.-7) Thus we an draw the following onlusion Group Veloity The group veloity of any partile (massive or massless) is equal to the derivative of its total relativisti energy with respet to its relativisti momentum.. Phase Veloity and Group Veloity as a Funtion of the de Broglie and the Compton Wavelengths of a Partile.1 Phase Veloity In this setion we shall derive the expression of the phase veloity of a partile as a funtion of its de Broglie wavelength and its Compton wavelength. To do that I will onsider equation (1.1-6) and the de Broglie law E p (.1-1) Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 3
4 p h λ (.1-) where h Plank's onstant λ de Broglie wavelength Now I shall define the Compton momentum, p C, as follows p C (.1-3) The Compton momentum is, as far as I know, not normally defined anywhere in the literature. However, sine the Compton momentum is a very important onept it is onvenient to introdue it here. This definition will allow us to write the Einstein's equation (1.-1) for the total relativisti energy of a partile, as follows E p + p C (.1-4) Taking as a ommon fator we an write E ( p + p C ) (.1-5) Before we ontinue I shall define the Compton wavelength of a partile of rest mass, as h p C h (.1-6) The Compton wavelength was introdued by the Amerian physiist Arthur Compton to explain the sattering of photons by eletrons. The Compton wavelength of a partile is the wavelength of a photon whose energy is equal to the rest energy, E 0, of the partile ( E 0 ). Now let us substitute the relativisti momentum p with the seond side of equation (.1-) and the Compton momentum with the Compton wavelength given by equation (.1-6). This gives ( E h λ + h ) λ (.1-7) C Taking h as a ommon fator we have E h ( 1 λ + 1 ) (.1-8) This equation an be written as E h λ ( 1 + λ ) (.1-9) Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 4
5 Taking the square root on both sides E h λ 1 + λ (.1-10) Now we use equation for the phase veloity: (1.1-6), where we substitute E with the seond side of the above equation, and p with the seond side of equation (.1-). These substitutions yield E p ( h λ 1 + λ ) λ h (.1-11) Finally, simplifying, we get the formula for the phase veloity in terms of the de Broglie wavelength, λ, and the Compton wavelength, 1 + λ (.1-1) Partiular ase of the phase veloity for photons For the partiular ase of photons, the rest mass is zero, mathematially therefore 0 (.1-13) h 1 0 (.1-14) (.1-15) (.1-16) Thus the phase veloity for photons equals the speed of light. We begin from equation (1.1-7). Group Veloity Solving for we get (.-1) (.-) Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 5
6 Now we substitute the denominator,, with the seond side of equation (.1-1) and after simplifying we get the formula for the group veloity in terms of the de Broglie wavelength, λ, and the Compton wavelength, 1 + λ (.-3) Partiular ase of the group veloity for photons For the partiular ase of photons, the rest mass is zero, mathematially this means that 0 (.-4) therefore (.-5) (.-6) Thus the group veloity for photons also equals the speed of light 3. Phase Veloity and Group Veloity as a Funtion of the Angular Frequeny and the Wave Number Let us multiply equation (.1-1) by ħ /ħ : 3.1 Phase Veloity ħ ħ E p E ħ ħ p (3.1-1) But we know that the total energy of a photon is proportional to its frequeny E ħ ω (3.-) and the momentum of a photon is proportional to its wave number p ħ k (3.-3) Thus equations (3.-) and (3.-3) allow us to rewrite equation (3.1-1) as ω k (3.-4) Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 6
7 Let us multiply equation (1.-7) by ħ /ħ : 3. Group Veloity ħ ħ de d (E /ħ) dp d ( p /ħ) (3.-1) Using equations (3.-) and (3.-3) from the previous subsetion; equation (3.-1) transforms into d ω dk (3.-) 3. Physial Meanings of the Compton Wavelength We have pointed out that the Compton wavelength of a partile is the wavelength of a photon whose energy is equal to the rest energy of the partile. There are, however, many other equivalent definitions. We an take equation (.-3), for example, and make the de Broglie wavelength of the partile the same as its Compton wavelength. Mathematially this means that we substitute λ with. This yields 1 + λ C (3-1) (3-) Thus the Compton wavelength of a partile is the wavelength assoiated with a partile that is moving at a speed equal to times the speed of light. Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 7
8 4. Summary The two following tables summarises the above results. Massive partiles (e.g. eletrons) Massless partiles (e.g. photons) Phase veloity (Speial Relativity) (See setion 1) E p Phase veloity (De Broglie) (See setion ) 1 + λ where h is the Compton wavelength Phase veloity (Angular frequeny and wave number) (See setion 3) ω k Table 1: Phase veloity formulas. Both massive partiles and photons obey the same equation. However for the latter the Compton wavelength is infinite. This means that, for photons, the phase veloity equal the speed of light in vauum. Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 8
9 Massive partiles (e.g. eletrons) Massless partiles (e.g. photons) Group veloity (Speial Relativity) (See setion 1) de dp Group veloity (De Broglie) (See setion ) 1 + λ where h is the Compton wavelength Group veloity (Angular frequeny and wave number) (See setion 3) d ω dk Table : Group veloity formulas. Both massive partiles and photons obey the same equation. However for the latter the Compton wavelength is infinite. This means that, for photons, the group veloity equal the speed of light in vauum. Two of the physial meanings of the Compton wavelength are Physial Meanings of the Compton Wavelength The Compton wavelength of a partile is 1) the wavelength of a photon whose energy is equal to the rest energy, E 0, of the partile ( E 0 ). ) the wavelength of a partile that is moving at a speed equal to 1/ times the speed of light. In other words Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 9
10 If then λ Phase Veloity and Group Veloity for Beginners - v3 Copyright Rodolfo A. Frino. All rights reserved. 1
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