Quantum Mechanics for Mathematicians

and intertwining operators

Ugr=eΓ

′BF(rμ)

=e

r 2 (a^2 −(a†)^2 )

which satisfy

Ugr

(

a

a†

)

Ug−r^1 =e

r



0 1

1 0



(

a

a†

)

=

(

coshr sinhr

sinhr coshr

)(

a

a†

)

(24.9)

The operator^12 (a^2 −(a†)^2 ) does not commute with the number operatorN,
or the harmonic oscillator HamiltonianH, so the transformationsUgrare not
“symmetry transformations”, preserving energy eigenspaces. In particular they
act non-trivially on the state| 0 〉, taking it to a different state

| 0 〉r=e

r 2 (a (^2) −(a†) (^2) )
| 0 〉

24.3 Implications of the choice ofz,z

The definition of annihilation and creation operators requires making a specific
choice, in our case

z=

1

√

2

(q+ip), z=

1

√

2

(q−ip)

for complexified coordinates on phase space, which after quantization becomes
the choice

a=

1

√

2

(Q+iP), a†=

1

√

2

(Q−iP)

Besides the complexification of coordinates on phase spaceM, the choice of
zintroduces a new piece of structure into the problem. In chapter 26 we’ll
examine other possible consistent such choices, here will just point out the
various different ways in which this extra structure appears.

• The Schr ̈odinger representation of the Heisenberg group comes with no
  particular distinguished state. The unitarily equivalent Bargmann-Fock
  representation does come with a distinguished state, the constant function
  
  
  1. It has zero eigenvalue for the number operatorN =a†a, so can be
     thought of as the state with zero “quanta”, or the “vacuum” state and
     can be written| 0 〉. Such a constant function could also be characterized
     (up to scalar multiplication), as the state that satisfies the condition
  
  
  
  

a| 0 〉= 0

• The choice of coordinateszandzgives a distinguished choice of Hamil-
  tonian function,h=zz. After quantization this corresponds to a distin-
  guished choice of Hamiltonian operator

H=

1

2

(a†a+aa†) =a†a+

1

2

=N+

1

2