3.6: Momentum Representation
( \newcommand{\kernel}{\mathrm{null}\,}\)
Fourier’s theorem (see Section [s2.9]), applied to one-dimensional wavefunctions, yields ψ(x,t)=1√2π∫∞−∞ˉψ(k,t)e+ikxdk,ˉψ(k,t)=1√2π∫∞−∞ψ(x,t)e−ikxdx,
At this stage, it is convenient to introduce a useful function called the Dirac delta-function . This function, denoted δ(x), was first devised by Paul Dirac , and has the following rather unusual properties: δ(x) is zero for x≠0, and is infinite at x=0. However, the singularity at x=0 is such that ∫∞−∞δ(x)dx=1.
Suppose that ψ(x)=δ(x−x0). It follows from Equations ([e3.65]) and ([e3.69]) that
ϕ(p)=e−ipx0/ℏ√2πℏ
Hence, Equation ([e3.64]) yields the important result δ(x−x0)=12πℏ∫∞−∞e+ip(x−x0)/ℏdp.
It turns out that we can just as easily formulate quantum mechanics using the momentum-space wavefunction, ϕ(p,t), as the real-space wavefunction, ψ(x,t). The former scheme is known as the momentum representation of quantum mechanics. In the momentum representation, wavefunctions are the Fourier transforms of the equivalent real-space wavefunctions, and dynamical variables are represented by different operators. Furthermore, by analogy with Equation ([e3.55]), the expectation value of some operator O(p) takes the form ⟨O⟩=∫∞−∞ϕ∗(p,t)O(p)ϕ(p,t)dp.
Consider momentum. We can write ⟨p⟩=∫∞−∞ψ∗(x,t)(−iℏ∂∂x)ψ(x,t)dx=12πℏ∫∞−∞∫∞−∞∫∞−∞ϕ∗(p′,t)ϕ(p,t)pe+i(p−p′)x/ℏdxdpdp′,
Consider displacement. We can write ⟨x⟩=∫∞−∞ψ∗(x,t)xψ(x,t)dx=12πℏ∫∞−∞∫∞−∞∫∞−∞ϕ∗(p′,t)ϕ(p,t)(−iℏ∂∂p)e+i(p−p′)x/ℏdxdpdp′.
Finally, let us consider the normalization of the momentum-space wavefunction ϕ(p,t). We have ∫∞−∞ψ∗(x,t)ψ(x,t)dx=12πℏ∫∞−∞∫∞−∞∫∞−∞ϕ∗(p′,t)ϕ(p,t)e+i(p−p′)x/ℏdxdpdp′.
The existence of the momentum representation illustrates an important point. Namely, there are many different, but entirely equivalent, ways of mathematically formulating quantum mechanics. For instance, it is also possible to represent wavefunctions as row and column vectors, and dynamical variables as matrices that act upon these vectors.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)