Re: Creation and Annihilation Operators in QM
- From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
- Date: Wed, 15 Nov 2006 23:38:07 +0000 (UTC)
Mike wrote:
Suppose a self-adjoint operator F on a Hilbert space has discrete
eigenvalues m_1 , m_2, m_3, etc. It is sometimes possible to find a
"creation operator" C such that if \phi is an eigenfunction of F for
the eigenvalue m_n, then C\phi is an eigenfunction with eigenvalue
m_{n+1}. Similarly an annihilation operator A does the opposite. Thus
if \phi has eigenvalue m_n then A\phi has eigenvalue m_{n-1}. The
reason for the terminology is that one thinks of C and A as creating or
annihilating quanta. Two classical examples of this in QM:
1) If L_z is the z-component of angular momentum (eigenvalues are hm
for m=0,1,2) then one can show that L_x + iL_y and L_x - iL_y are
creation and annihilation operators respectively.
2) For the harmonic oscillator with potential kx^2 one can show that C
= cx + i dp and A = cx - dp are creation and annihilation operators for
the Hamiltonian (x is position, p is momentum, and c and d are suitable
constants. This is the most elegant way to mathematically derive the
spectrum of energy eigenvalues.
Question: Is there any theorem along the following lines. If an
operator F has discrete spectrum and satisfies suitable hypotheses,
then there exist creation and annihilation operators for F? I am no
expert in functional analysis, but it seems to me that such a theorem
should be possible.
Yes. This is easy to see if the operators are represented
in the basis of eigenvectors. Any assignment of orthogonal vectors g_k
of norm <=1 to the eigenvectors f_k extends to a linear operator F,
since the general vector f has an expansion
f=sum c_k f_k with sum |c_k|^2 < inf,
and
Ff:= sum c_k g_k
produces a well-defined Ff. Clearly, the mapping is linear.
Arnold Neumaier
.
- References:
- Creation and Annihilation Operators in QM
- From: Mike
- Creation and Annihilation Operators in QM
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