3.5: Operators
( \newcommand{\kernel}{\mathrm{null}\,}\)
An operator, O (say), is a mathematical entity that transforms one function into another: that is,
O(f(x))→g(x).
xdfdx≠ddx(xf).
xx2=x2x.
Now, from Equations ([e3.22]) and ([e3.38]),
⟨x⟩=∫∞−∞ψ∗xψdx,⟨p⟩=∫∞−∞ψ∗(−iℏ∂∂x)ψdx.
These expressions suggest a number of things. First, classical dynamical variables, such as x and p, are represented in quantum mechanics by linear operators that act on the wavefunction. Second, displacement is represented by the algebraic operator x, and momentum by the differential operator −iℏ∂/∂x: that is, \[\label{e3.54} p \equiv -{\rm i}\,\hbar\,\frac{\partial}{\partial x}.\]
Finally, the expectation value of some dynamical variable represented by the operator O(x) is simply ⟨O⟩=∫∞−∞ψ∗(x,t)O(x)ψ(x,t)dx.
Clearly, if an operator is to represent a dynamical variable that has physical significance then its expectation value must be real. In other words, if the operator O represents a physical variable then we require that ⟨O⟩=⟨O⟩∗, or ∫∞−∞ψ∗(Oψ)dx=∫∞−∞(Oψ)∗ψdx,
where O∗ is the complex conjugate of O. An operator that satisfies the previous constraint is called an Hermitian operator. It is easily demonstrated that x and p are both Hermitian. The Hermitian conjugate, O†, of a general operator, O, is defined as follows:
Suppose that we wish to find the operator that corresponds to the classical dynamical variable xp. In classical mechanics, there is no difference between xp and px. However, in quantum mechanics, we have already seen that xp≠px. So, should we choose xp or px? Actually, neither of these combinations is Hermitian. However, (1/2)[xp+(xp)†] is Hermitian. Moreover, (1/2)[xp+(xp)†]=(1/2)(xp+p†x†)=(1/2)(xp+px), which neatly resolves our problem of the order in which to place x and p.
It is a reasonable guess that the operator corresponding to energy (which is called the Hamiltonian, and conventionally denoted H) takes the form H≡p22m+V(x).
Finally, if O(x,p,E) is a classical dynamical variable that is a function of displacement, momentum, and energy then a reasonable guess for the corresponding operator in quantum mechanics is (1/2)[O(x,p,H)+O†(x,p,H)], where p=−iℏ∂/∂x, and H=iℏ∂/∂t.
Contributors and Attributions
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)