4.3: Normalization of the Wavefunction
( \newcommand{\kernel}{\mathrm{null}\,}\)
Now, a probability is a real number between 0 and 1. An outcome of a measurement which has a probability 0 is an impossible outcome, whereas an outcome which has a probability 1 is a certain outcome. According to Eq. 1.2.2 , the probability of a measurement of x yielding a result between −∞ and +∞ is is
Px∈−∞:∞(t)=∫∞−∞|ψ(x,t)|2dx
However, a measurement of x must yield a value between −∞ and +∞, since the particle has to be located somewhere. It follows that Px∈−∞:∞=1, or
∫∞−∞|ψ(x,t)|2dx=1
which is generally known as the normalization condition for the wavefunction.
For example, suppose that we wish to normalize the wavefunction of a Gaussian wave packet, centered on x=x0 and of characteristic width σ
ψ(x)=ψ0e−(x−x0)2/(4σ2)
In order to determine the normalization constant ψ0, we simply substitute Eq. 1.3.3 and into Eq. 1.3.2, to obtain
|ψ0|2∫∞−∞e−(x−x0)2/(2σ2)dx=1
Changing the variable of integration to y=(x−x0)/(√2σ), we get
|ψ0|2√2σ∫∞−∞e−y2dy=1
However,
∫∞−∞e−y2dy=√π
which implies that
|ψ0|2=1(2πσ2)1/2
Hence, a general normalized Gaussian wavefunction takes the form
ψ(x)=eiφ(2πσ2)1/4e−(x−x0)2/(4σ2)
where φ is an arbitrary real phase-angle.
Now, it is important to demonstrate that if a wave function 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 wave function is untenable, since 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
for wavefunctions satisfying Schrödinger's equation. The above equation gives
ddt∫∞−∞ψ∗ψdx=∫∞−∞(∂ψ∗∂tψ+ψ∗∂ψ∂t)dx=0
Now, multiplying Schrödinger's equation by ψ∗/(iℏ), we obtain
ψ∗∂ψ∂t=iℏ2mψ∗∂2ψ∂x2−iℏV|ψ|2
The complex conjugate of this expression yields
ψ∂ψ∗∂t=−iℏ2mψ∂2ψ∗∂x2+iℏV|ψ|2
[since (AB)∗=A∗B∗,A∗∗=A, and i∗=−i]. Summing the previous two equations, we get
∂ψ∗∂tψ+ψ∗∂ψ∂t=iℏ2m(ψ∗∂2ψ∂x2−ψ∂2ψ∗∂x2)=iℏ2m∂∂x(ψ∗∂ψ∂x−ψ∂ψ∗∂x)
Equations 1.3.10 and 1.3.13 can be combined to produce
ddt∫∞−∞|ψ|2dx=iℏ2m[ψ∗∂ψ∂x−ψ∂ψ∗∂x]∞−∞=0
The above equation is satisfied provided
|ψ|→0 as |x|→∞
However, this is a necessary condition for the integral on the left-hand side of Eq. 1.3.2 to converge. Hence, we conclude that all wavefunctions which are square-integrable [i.e., are such that the integral in Eq. 1.3.2 converges] have the property that if the normalization condition (1.3.2) is satisfied at one instant in time then it is satisfied at all subsequent times.
It is also possible to demonstrate, via very similar analysis to the above, that
dPx∈a:bdt+j(b,t)−j(a,t)=0
where Px∈a:b is defined in Eq. 1.2.2, and
j(x,t)=iℏ2m(ψ∂ψ∗∂x−ψ∗∂ψ∂x)
is known as the probability current. Note that j is real. Equation 1.3.16 is a probability conservation equation. According to this equation, the probability of a measurement of x lying in the interval a to b evolves in time due to the difference between the flux of probability into the interval [i.e., j(a,t)], and that out of the interval [i.e., j(b,t)]. Here, we are interpreting j(x,t) as the flux of probability in the +x -direction at position x and time t.
Note, finally, that not all wavefunctions can be normalized according to the scheme set out in Eq. 1.3.2. For instance, a plane wave wavefunction
ψ(x,t)=ψ0ei(kx−ωt)
is not square-integrable, and, thus, cannot be normalized. For such wavefunctions, the best we can say is that
Px∈a:b(t)∝∫ba|ψ(x,t)|2dx
In the following, all wavefunctions are assumed to be square-integrable and normalized, unless otherwise stated.
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)