6.1 Derivation of the Schr¨odinger Wave Equation
6.1.1 The Time Dependent Schr¨odinger Wave Equation
In the discussion of the particle in an inﬁnite potential well, it was observed that the wave function of a particle of ﬁxed energy E could most naturally be written as a linear combination of wave functions of the form
Ψ(x,t) = Aei(kx−ωt) (6.1)
representing a wave travelling in the positive x direction, and a corresponding wave travelling in the opposite direction, so giving rise to a standing wave, this being necessary in order to satisfy the boundary conditions. This corresponds intuitively to our classical notion of a particle bouncing back and forth between the walls of the potential well, which suggests that we adopt the wave function above as being the appropriate wave function
Chapter 6 The Schr¨odinger Wave Equation 43
for a free particle of momentum p = !k and energy E = !ω. With this in mind, we can then note that ∂2Ψ ∂x2 =−k2Ψ (6.2) which can be written, using E = p2/2m = !2k2/2m:
= p2 2m
which can be written, using E = !ω: i!∂Ψ ∂t
= !ωψ = EΨ. (6.5)
We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then E = p2/2m+V(x) so that EΨ = p2 2m Ψ+V(x)Ψ (6.6) where Ψ is now the wave function of a particle moving in the presence of a potential V(x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have − !2 2m ∂2ψ ∂x2 +V(x)Ψ = i!∂ψ ∂t (6.7) which is the famous time dependent Schr¨odinger wave equation.
Curiously enough, to answer this question requires ‘extracting’ the time dependence from the time dependent Schr¨odinger equation. To see how this is done, and its consequences, we will turn our attention to the closely related time independent version of this equation.