Representation of d’Alembertian Operator in Quantum Mechanics
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IOSR Journal of Applied Physics (IOSR-JAP) vol.7 issue.1 version.2
=0.
V. Conclusion
Finally the d’Alembertian operator takes the form in quantum representation as
□2
= ħ2
/4m2
c2
.Δ4
- {V(r)/mc2
-1} Δ2
+V2
(r)/ħ2
c2
.
As well as in this respect, the wave equation in quantum mechanical interpretation is
[ħ2
/4m2
c2
.Δ4
- {V(r)/mc2
-1} Δ2
+V2
(r)/ħ2
c2
](r,t)=0.](https://image.slidesharecdn.com/m07128081-151121103456-lva1-app6891/85/Representation-of-d-Alembertian-Operator-in-Quantum-Mechanics-1-320.jpg)
![Representation of d’Alembertian Operator in Quantum Mechanics
DOI: 10.9790/4861-07128081 www.iosrjournals.org 81 | Page
References
[1]. J.D.Jackson, Classical Electrodynamics.
[2]. E.P.Ney, Electromagnetism and Relativity.
[3]. P.A.M.Dirac, Quantum Mechanics.
[4]. V.Rojansky, Introductory Quantum Mechanics.](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Representation of d’Alembertian Operator in Quantum Mechanics
- 1. IOSR Journal of Applied Physics (IOSR-JAP) e-ISSN: 2278-4861.Volume 7, Issue 1 Ver. II (Jan.-Feb. 2015), PP 80-81 www.iosrjournals.org DOI: 10.9790/4861-07128081 www.iosrjournals.org 80 | Page Representation of d’Alembertian Operator in Quantum Mechanics Umasankar Dolai Assistant teacher, Garhbeta South C.L.R.C., Satbankura-721253, Dist.-Paschim Medinipur, West Bengal , India Abstract: The quantum mechanical formulation of d’Alembertian operator □2 will be drawn in this paper. Keywords: Classical Mechanics, Quantum Mechanics, General and Operator Algebra. I. Introduction In classical mechanics, the d’Alembertian operator □2 =Δ2 -1/c2 .δ2 /δt2 ; where Δ2 = δ2 /δx2 + δ2 /δy2 + δ2 /δz2 . Now what will be the form of □2 in quantum representation? II. d’Alembertian Operator In quantum mechanics, the momentum operator p=ħ/i.Δ and the energy operator E=iħ.δ/δt. Hence p2 = ħ2 /i2 .Δ2 =-ħ2 Δ2 and E2 =i2 ħ2 .δ2 /δt2 =- ħ2 .δ2 /δt2 Now Δ2 =-p2 /ħ2 and δ2 /δt2 =-E2 /ħ2 Thus Δ2 -1/c2 .δ2 /δt2 = E2 /c2 ħ2 -p2 /ħ2 Therefore □2 =(E2 /c2 -p2 )/ħ2 III. Energy and Momentum It is known that E=p2 /2m+V(r). As p=ħ/i.Δ; so E2 ={-ħ2 /2m.Δ2 +V(r)}2 =ħ4 /4m2 .Δ4 -2. ħ2 /2m.Δ2 .V(r)+V2 (r) Hence E2 /c2 = ħ4 /4m2 c2 .Δ4 -ħ2 /mc2 .V(r). Δ2 +V2 (r)/c2 Thus E2 /c2 -p2 = ħ4 /4m2 c2 .Δ4 -ħ2 /mc2 .V(r). Δ2 +V2 (r)/c2 +ħ2 Δ2 = ħ4 /4m2 c2 .Δ4 -ħ2 Δ2 {V(r)/mc2 -1}+V2 (r)/c2 Therefore (E2 /c2 -p2 )/ħ2 = ħ2 /4m2 c2 .Δ4 - {V(r)/mc2 -1} Δ2 +V2 (r)/ħ2 c2 IV. Wave Equation The equation of motion of wave propagation is Δ2 (r,t)=1/u2 .δ2 /δt2 .(r,t). If u=c, then { Δ2 -1/c2 .δ2 /δt2 }(r,t)=0. Again from the above discussion, (E2 /c2 -p2 )/ħ2 = □2 = ħ2 /4m2 c2 .Δ4 - {V(r)/mc2 -1} Δ2 +V2 (r)/ħ2 c2 . Therefore □2 (r,t)= [ħ2 /4m2 c2 .Δ4 - {V(r)/mc2 -1} Δ2 +V2 (r)/ħ2 c2 ](r,t)=0. V. Conclusion Finally the d’Alembertian operator takes the form in quantum representation as □2 = ħ2 /4m2 c2 .Δ4 - {V(r)/mc2 -1} Δ2 +V2 (r)/ħ2 c2 . As well as in this respect, the wave equation in quantum mechanical interpretation is [ħ2 /4m2 c2 .Δ4 - {V(r)/mc2 -1} Δ2 +V2 (r)/ħ2 c2 ](r,t)=0.
- 2. Representation of d’Alembertian Operator in Quantum Mechanics DOI: 10.9790/4861-07128081 www.iosrjournals.org 81 | Page References [1]. J.D.Jackson, Classical Electrodynamics. [2]. E.P.Ney, Electromagnetism and Relativity. [3]. P.A.M.Dirac, Quantum Mechanics. [4]. V.Rojansky, Introductory Quantum Mechanics.