EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM (DIRAC EQ.)

Transcription

1 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. ABSTRACT The charge of the electron and the proton is assumed to be distributed in space. The potential energy of a specific charge distribution is determined. Perturbation theory is used to calculate the shift in the S energy level of the hydrogen atom due to the proton and electron size. I. SOLUTION TO THE DIRAC EQUATION The potential energy of a point proton and a point electron at rest is V r = e 2 /r. When this potential energy is put into the Dirac equation, the wave function for the S energy level is gρy θ, φ ψ n,l,j ρ, θ, φ = ψ,,/2 ρ, θ, φ = ifρ 3 Y θ, φ 2 ifρ 3 Y θ, φ Date: December, 25.

2 2 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. where Y l,m is a spherical harmonic 2, ρ = 2r/a, a is the Bohr radius, 2 3/2 + ɛ gρ = a 2Γ2γ + exp ρ/2ργ, 2 fρ = ɛgρ/ + ɛ, γ = α 2, the fine structure constant α = e 2 / c, ɛ = E/mc 2, and Γ refers to the gamma function. II. THE POTENTIAL ENERGY Take the proton charge to be uniformly distributed on a spherical shell of radius r in its rest frame. The potential energy of this proton and a point electron is V r = e2 r Hr r + e2 r Hr r 3 where H is the unit step function. Subtract e 2 /r from V r to get the perturbing potential energy, δv r, due to the proton size. The result is δv r = e2 r Hr r + +e2 r Hr r. 4 III. SHIFT OF THE S ENERGY LEVEL DUE TO PROTON SIZE The energy shift of the S energy level due to proton size is δe = ψ,,/2 r, θ, φδv rψ,,/2r, θ, φ d 3 r. 5

3 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. 3 Since the wave function was given in terms of ρ and not r, rewrite the above equation as δe = ψ,,/2 ρ, θ, φδvρψ,,/2ρ, θ, φ a 2 3 d 3 ρ 6 where d 3 ρ = ρ 2 sinθdθ dφ dρ = ρ 2 dρ dω, ρ = 2r /a and δv ρ = 2e2 + Hρ ρ. 7 a ρ ρ Note that ψ,,/2 ρ, θ, φψ,,/2ρ, θ, φ sinθdθ dφ = [ gρ 2 Yθ, 2 φ + f 2 ρ 3 Y θ, 2 φ Y θ, φy θ, φ ]dω = gρ 2 2 ɛ 8 since the spherical harmonics are normalized. So δe = 2e 2 ρ exp ρρ 2γ + dρ. 9 a Γ2γ + ρ ρ Taylor expand the exponential dropping terms of the order ρ squared and higher. Note ρ = 2r /a <<. Then δe = 2e 2 [ a Γ2γ + ρ ρ 2γ ρ 2γ+ dρ ρ ] + ρ 2γ ρ 2γ dρ. ρ After integration and a little algebra δe = 2e 2 [ + a Γ2γ + ρ 2γ 2γ2γ + ρ 2γ+ 2γ + 2γ + 2 ].

4 4 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. Substitute ρ = 2r /a, and get δe = 2e 2 2r 2γ [ 2r γ ], 2 a Γ2γ + a 2γ2γ + a γ + where γ = α 2 since α 37. Set γ =, and find δe e2 2r 2 a 3a 2 r. 3 a IV. ELECTRON SIZE INCLUDED IN THE POTENTIAL ENERGY Again take the proton charge to be uniformly distributed on a spherical shell of radius r in the proton rest frame, and take the electron charge to be uniformly distributed on a spherical shell of radius a in the electron rest frame. The potential energy of the proton and electron is 3 V e r = e2 r Hr a r e2 r Hr r a+[v i r+v o r][hr+a r Hr r a] 4 where V i r = e2 2r + r2 a2 r 2 2ra, 5 and V o r = e2 r + a r. 6 2ra Subtract e 2 /r from V e r to get the perturbing potential energy δv e r = e2 r Hr a r+ +e2 r Hr +a r+[v i r+v o r][hr+a r Hr r a]. 7

5 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. 5 Express δv e r in terms of ρ = 2r/a, ρ = 2r /a, and ā = 2a/a, and find δv e ρ = 2e2 a ρ Hρ ā ρ+ 2e2 a ρ Hρ +ā ρ+[v i ρ+v o ρ][hρ+ā ρ Hρ ρ ā] 8 where V i ρ = 2e2 2ρ a + ρ2 ā2 ρ 2 2ρa, 9 and V o ρ = 2e2 ρ + ā ρ. 2 2ρa ā Substitute δv e ρ into Eq. 6. The resulting energy shift of the S energy level is δe = δe + δe 2 + δe 3 where δe = 2e 2 a Γ2γ + ρ ρ ā exp ρρ 2γ dρ, 2 δe 2 = 2e 2 a Γ2γ + ρ +ā exp ρρ 2γ dρ, 22 and δe 3 = Γ2γ + ρ +ā ρ ā exp ρρ 2γ [V i ρ + V o ρ]dρ. 23 Taylor expand the exponential, and drop higher order terms in ρ. Then δe = 2e 2 a Γ2γ + ρ ρ ā ρ 2γ ρ 2γ+ dρ, 24 δe 2 = 2e 2 a Γ2γ + ρ +ā ρ 2γ ρ 2γ dρ, 25

6 6 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. and δe 3 = Γ2γ + ρ +ā ρ ā After integration, δe can be put in the form ρ 2γ ρ 2γ+ [V i ρ + V o ρ]dρ. 26 δe = 2e 2 [ ρ 2γ+ ā 2γ+ ρ 2γ+2 ā 2γ+2]. 27 a Γ2γ + ρ 2γ + ρ 2γ + 2 ρ Taylor expand the two terms in parentheses dropping higher order terms in ā/ρ, and find δe = 2e 2 a Γ2γ + [ ρ 2γ Similarly after integration, δe 2 = 2γ + ā + 2γā2 ρ 2ρ 2 ρ 2γ+ 2γ + 2 ā + 2γ + ā2 ]. 28 ρ 2ρ 2 2e 2 [ ρ 2γ ā 2γ ρ 2γ+ ā 2γ+. 29 a Γ2γ + 2γ ρ 2γ + ρ Again Taylor expand dropping higher powers of ā/ρ, and find δe 2 = +2e 2 [ ρ 2γ a Γ2γ + 2γ + ā 2γ ā2 + ρ 2ρ 2 ρ 2γ+ 2γ + + ā + 2γā2 ]. 3 ρ 2ρ 2 Note that δe and δe 2 can be added to yield δe + δe 2 = +2e 2 a Γ2γ + [ ρ 2γ 2γ + 2γ + 2ā â2 ρ 2ρ 2 ρ 2γ+ 2γ + 22γ + 2â + ā2 ]. 3 ρ 2ρ 2

7 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. 7 It is convenient to set γ = now. Then δe + δe 2 = +2e2 [ ρ 2 a ā â2 ρ 2ρ 2 + ρ 3 2 2â + ā2 ]. 32 ρ 2ρ 2 Next substitute ρ = 2r /a and ā = 2a/a in Eq. 32, and get δe +δe 2 = +e2 [ 2r 2 a 3a 2 r a δe 3 = + 8r a a 2 2r a Finally substitute V i ρ and V o ρ in Eq. 26, and get 2e 2 a Γ2γ + ρ +ā ρ ā [ ρ 2γ ρ 2γ+][ + ρ2 ā 2 ρ 2 2ρ 2ρā 2a2 a 2 2r ]. 33 a + ρ + ā ρ 2ρā To avoid a lot of algebra, δe 3 will be approximated using the mean value theorem for integrals. Since ρ is in the middle of the limits of integration, set ρ = ρ in the integrand. Then δe 3 = 34 2e2 2ā [ a Γ2γ + ρ2γ ρ2γ+ 2 ā ]. 35 2ρ 2ρ ] dρ. Again set γ =, and find δe 3 = 2e2 2ā [ ρ ρ ā ] a 2! 4 ρ. 36 Substitute ρ = 2r /a and ā = 2a/a in Eq. 36, and get δe 3 = e2 [ 8ar a a 2 2r a Add the results, and find 8a2 4a 2 2r ]. 37 a δe = δe + δe 2 + δe 3 = e2 2r 2 a 3a 2 r. 38 a

8 8 EFFECT OF PARTICLE SIZE ON H-ATOM SPECTRUM DIRAC EQ. Terms like r a/a 2 and a2 /a 2 do not appear. Presumably the electron radius will appear as a 3 /a 3 when more terms are kept in the Taylor expansion of the exponential. However a 3 /a 3 is quite small. ACKNOWLEDGEMENTS I thank Ben for his many improvements to the paper. References Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One and Two- Electron AtomsSpringer-Verlag, Berlin, 957, p Hans A. Bethe and Edwin E. Salpeter, Quantum Mechanics of One and Two- Electron AtomsSpringer-Verlag, Berlin, 957, p of Electron and Proton Size On Spin-Orbit Coupling