wave equation of a particle in a scalar potential

In order to form the wave equation of a particle in a potential V (r), we operate at first under the conditions of the ‘geometrical optics approximation’ and seek to form an equation of propagation for a wave packet Ψ(r,t) moving in accordance with the de Broglie theory.

The center of the packet travels like a classical particle whose position, momentum, and energy we shall designate by r_cl., p_cl., and E_cl., respectively. These quantities are connected by the relation

p2cl. Ecl.= H (rcl.,pcl.) = ---+ V(rcl.) 2m

(1)

H(r_cl.,p_cl.) is the classical Hamiltonian. We suppose that V (r) does not depend upon the time explicitly (conservative system), although this condition is not absolutely necessary for the present argument to hold. Consequently E_cl. remains constant in time, while r_cl. and p_cl. are well-defined functions of t. Under the approximate conditions considered here, V (r) remains practically constant over a region of the order of the size of the wave packet; therefore

V (r)Ψ (r, t) ≈ V (r )Ψ(r,t) cl.

(2)

On the other hand, if we restrict ourselves to time intervals sufficiently short so that the relative variation of p_cl. remains negligible, Ψ(r,t) may be considered as a superposition of plane waves of the type

∫ Ψ (r,t) = F (p) expi(p⋅r− Et)∕ℏdp

(3)

whose frequencies are in the neighborhood of E_cl.∕ℏ and whose wave vectors lie close to p_cl.∕ℏ. Therefore

iℏ ∂-Ψ (r,t) ≈ E Ψ (r,t) ∂t cl.

ℏ --∇ Ψ(r,t) ≈ pcl.(t)Ψ (r,t) i

(4)

and taking the divergence of this last express ion, one obtains

− ℏ2∇2 Ψ(r,t) ≈ p2cl.Ψ (r,t)

(5)

combining the relations (2),(3), and (4) and making use of equation (1), we obtain

( ) ∂ ℏ2 2 p2cl. ıℏ∂t-Ψ + 2m--∇ Ψ − V Ψ ≈ Ecl.− 2m-− V (rcl.) Ψ ≈ 0

The wave packet Ψ(r,t) satisfies - at least approximately - a wave equation of the type we are looking for. We are very naturally led to adopt this equation as the wave equation of a particle in a potential, and we postulate that in all generality, even when the conditions for the ‘geometrical optics’ approximation are not fulfilled, the wave Ψ satisfies the equation

∂ ( ℏ2 ) iℏ--Ψ (r,t) = − ----∇2 + V (r) Ψ (r,t) ∂t 2m

(6)

It is the Schrödinger equation for a particle in a potential V (r).

[1] Messiah, Albert. ”Quantum mechanics: volume I.” Amsterdam, North-Holland Pub. Co.; New York, Interscience Publishers, 1961-62.

This entry is a derivative of the Public domain work [1].

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