37.2 Quadratic operators and dynamics

37.2 Quadratic operators and dynamics

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Other observables can be defined simply in terms of the field operators. These include (note that in all cases these formulas require interpretation as limits of finite sums in the finite cutof theory):

• The number operator $\hat{N}$ . A number density operator can be defined by

$$\hat{n}(x)={\hat{\Psi }}^{†}(x)\hat{\Psi }(x)$$

and integrated to get an operator with eigenvalues the total number of particles in a state

$$\begin{array}{rl}\hat{N}& ={\int }_{-\infty }^{\infty }\hat{n}(x)dx\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-i{p}^{\prime }x}{a}^{†}(p^{\prime })\frac{1}{\sqrt{2\pi }}{e}^{ipx}a(p)dpd{p}^{\prime }dx\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\delta (p-p^{\prime })a^{†}(p^{\prime })a(p)dpd{p}^{\prime }\\ & ={\int }_{-\infty }^{\infty }{a}^{†}(p)a(p)dp\end{array}$$

• The total momentum operator $\hat{P}.$ This can be defined in terms of field

operators as

$$\begin{array}{rl}\hat{P}& ={\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x)(-i\frac{d}{dx}\hat{\Psi }(x))dx\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-i{p}^{\prime }x}{a}^{†}(p^{\prime })(-i)(ip)\frac{1}{\sqrt{2\pi }}{e}^{ipx}a(p)dpd{p}^{\prime }dx\\ & ={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\delta (p-p^{\prime })p{a}^{†}(p^{\prime })a(p)dpd{p}^{\prime }\\ & ={\int }_{-\infty }^{\infty }p{a}^{†}(p)a(p)dp\end{array}$$

For more discussion of this operator and its relation to spatial translations, see section 38.3.1.

• The Hamiltonian $\hat{H}$ . As an operator quadratic in the field operators, this can be chosen to be

$$\hat{H}={\int }_{-\infty }^{\infty }{\hat{\Psi }}^{†}(x)\left(-\frac{1}{2m}\frac{d^2}{d{x}^2}\right)\hat{\Psi }(x)dx={\int }_{-\infty }^{\infty }\frac{p^2}{2m}{a}^{†}(p)a(p)dp$$

The dynamics of a quantum field theory is usually described in the Heisenberg picture, with the evolution of the field operators given by Fourier transformed versions of the discussion in terms of $a(p),a^{†}(p)$ of section 36.5. The quantum fields satisfy the general dynamical equation

$$\frac{d}{dt}{\hat{\Psi }}^{†}(x,t)=-i[{\hat{\Psi }}^{†}(x,t),\hat{H}]$$

which in this case is

$$\frac{\partial }{\partial t}{\hat{\Psi }}^{†}(x,t)=-\frac{i}{2m}\frac{{\partial }^2}{\partial x^2}{\hat{\Psi }}^{†}(x,t)$$

Note that the field operator ${\hat{\Psi }}^{†}(x,t)$ satisfies the (conjugate) Schr¨odinger equation, which now appears as a diferential equation for distributional operators rather than for wavefunctions. Such a diferential equation can be solved just as for wavefunctions, by Fourier transforming and turning diferentiation into multiplication, and we find

$${\hat{\Psi }}^{†}(x,t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-ipx}{e}^{i\frac{p^2}{2m}t}{a}^{†}(p)dp$$

Just as in the case of 36.5, this formal calculation involving the quantum field operators has an analog in terms of the $\Psi (x)$ and a quadratic function on the phase space. One can write

$$h={\int }_{-\infty }^{+\infty }\Psi (x)\frac{-1}{2m}\frac{{\partial }^2}{\partial x^2}\overline{\Psi }(x)dx$$

and the dynamical equations as

$$\frac{d}{dt}\Psi (x,t)=\{\Psi (x,t),h\}$$

which can be evaluated to give

$$\frac{\partial }{\partial t}\Psi (x,t)=-\frac{i}{2m}\frac{{\partial }^2}{\partial x^2}\Psi (x,t)$$

Note that there are other possible forms of the Hamiltonian function that give the same dynamics, related to the one we chose by integration by parts, in particular

$$\Psi (x)\frac{d^2}{d{x}^2}\overline{\Psi }(x)=\frac{d}{dx}\left(\Psi (x)\frac{d}{dx}\overline{\Psi }(x)\right)-|\frac{d}{dx}\Psi (x){|}^2$$

or

$$\Psi (x)\frac{d^2}{d{x}^2}\overline{\Psi }(x)=\frac{d}{dx}\left(\Psi (x)\frac{d}{dx}\overline{\Psi }(x)-\left(\frac{d}{dx}\Psi (x)\right)\overline{\Psi }(x)\right)+\left(\frac{d^2}{d{x}^2}\Psi (x)\right)\overline{\Psi }(x)$$

Neglecting integrals of derivatives (assuming boundary terms go to zero at infinity), one could have used

$$h=\frac{1}{2m}{\int }_{-\infty }^{+\infty }|\frac{d}{dx}\Psi (x){|}^{2}dx\mathrm{or}h=-\frac{1}{2m}{\int }_{-\infty }^{+\infty }\left(\frac{d^2}{d{x}^2}\Psi (x)\right)\overline{\Psi }(x)dx$$

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