3.2: Normalization of the Wavefunction
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Now, a probability is a real number lying between 0 and 1. An outcome of a measurement that has a probability 0 is an impossible outcome, whereas an outcome that has a probability 1 is a certain outcome. According to Equation ([e3.2]), the probability of a measurement of x yielding a result lying between −∞ and +∞ is Px∈−∞:∞(t)=∫∞−∞|ψ(x,t)|2dx.
For example, suppose that we wish to normalize the wavefunction of a Gaussian wave-packet, centered on x=x0, and of characteristic width σ (see Section [s2.9]): that is, ψ(x)=ψ0e−(x−x0)2/(4σ2).
Hence, a general normalized Gaussian wavefunction takes the form
ψ(x)=ei φ(2π σ2)1/4e−(x−x0)2/(4σ2),
It is important to demonstrate that if a wavefunction is initially normalized then it stays normalized as it evolves in time according to Schrödinger’s equation. If this is not the case then the probability interpretation of the wavefunction is untenable, because it does not make sense for the probability that a measurement of x yields any possible outcome (which is, manifestly, unity) to change in time. Hence, we require that ddt∫∞−∞|ψ(x,t)|2dx=0,
ddt∫∞−∞ψ∗ψdx=∫∞−∞(∂ψ∗∂tψ+ψ∗∂ψ∂t)dx=0.
ψ∗ ∂ψ∂t=i ℏ2 m ψ∗ ∂2ψ∂x2−iℏV|ψ|2.
The complex conjugate of this expression yields
ψ ∂ψ∗∂t=−i ℏ2 mψ ∂2ψ∗∂x2+iℏ V |ψ|2
[because (AB)∗=A∗B∗, A∗∗=A, and i∗=−i].
Summing the previous two equations, we get
∂ψ∗∂tψ+ψ∗∂ψ∂t=iℏ2 m(ψ∗∂2ψ∂x2−ψ∂2ψ∗∂t2)=iℏ2 m∂∂x(ψ∗∂ψ∂x−ψ∂ψ∗∂x).
Equations ([e3.12]) and ([e3.15]) can be combined to produce ddt∫∞−∞|ψ|2dx=iℏ2m[ψ∗∂ψ∂x−ψ∂ψ∗∂x]∞−∞=0.
It is also possible to demonstrate, via very similar analysis to that just described, that
Note, finally, that not all wavefunctions can be normalized according to the scheme set out in Equation ([e3.4]). For instance, a plane-wave wavefunction ψ(x,t)=ψ0ei(kx−ωt)
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)