Suppose we have the Hilbert space L^{2}(R^{n}) and we have n operators Q_{i} and n operators P_{i} defined in the usual way by:

Q_{i} ψ(q_{1},q_{2},...,q_{n}) = q_{i} ψ(q_{1},q_{2},...,q_{n})

P_{i} ψ(q_{1},q_{2},...,q_{n}) = -i $\frac{\partial}{\partial q_i}$ ψ(q_{1},q_{2},...,q_{n})

for i=1,2,...,n. The Q's are the position operators and P's the momentum operators of usual quantum mechanics.

These each define self-adjoint operators (when restricted to the appropriate domains).

You can look up in a text book the spectral decomposition of the Q_{i} to find the spectral family of projection operators E^{(i)}_{x} is defined by

E^{(i)}_{x} ψ(q_{1},q_{2},...,q_{n}) = ψ(q_{1},q_{2},...,q_{n}) if q_{i} < x
and
E^{(i)}_{x} ψ(q_{1},q_{2},...,q_{n}) = 0
if q_{i} >= x

We then have

< ψ_{1}, Q_{i} ψ_{2} > = $\int_{-\infty}^\infty$ x d < ψ_{1}, E^{(i)}_{x} ψ_{2} >

where <.,.> denotes the inner product in L^{2}(R^{n}) and the integral is a Riemann-Stieltjes integral.

The spectral family for the P_{i} is related to that for the Q's through the n-dimensional Fourier transform F. We have

< ψ_{1}, P_{i} ψ_{2} > = $\int_{-\infty}^\infty$ x d < ψ_{1}, F^{-1} E^{(i)}_{x} F ψ_{2} >
= $\int_{-\infty}^\infty$ x d < F ψ_{1}, E^{(i)}_{x} F ψ_{2} >

The operators Q_{i} and P_{i} are very well understood and appear in lots of textbooks. My question concerns taking *linear combinations* of them.

We define φ = ∑_{i=1..n} a_{i} Q_{i} + b_{i} P_{i}
for real coefficients a_{i} and b_{i}. This operator is self-adjoint so should also have a spectral family that defines its spectral decomposition.

The question is what is the spectral family of φ? Is it related to the spectral families of the Q's and P's? What are the projection operators E_{x} that allow us to write

< ψ_{1}, φ ψ_{2} > = $\int_{-\infty}^\infty$ x d < ψ_{1}, E_{x} ψ_{2} > ?

I'm interested in this because I'm interested in defining functions of the φ operator (for example defining the operator exp(iφ^{2}) or exp(iφ^{4}). The only way I know how to do this is using the functional calculus defined through the spectral decomposition of φ. If this decomposition was known then we'd have (for example):

< ψ_{1}, exp(iφ^{4}) ψ_{2} > = $\int_{-\infty}^\infty$ exp(i x ^{4}) d < ψ_{1}, E_{x} ψ_{2} >

In any event I want to know how to calculate < ψ_{1}, exp(iφ^{n}) ψ_{2} > say. i.e. I want a closed-form expression for what this number is.

If anyone knows how to do this or whether it's possible/impossible I'd really like to know.

PS - A good reference I've been using for this is "Linear Operators for Quantum Mechanics" by Thomas F. Jordan esp. chapter 3.