Download Lecture I: Dirac Notation

Lecture I: Dirac Notation
•
To describe a physical system, QM assigns a
complex number (`amplitude’) to each distinct
available physical state.
– (Or alternately: two real numbers)
– What is a `distinct physical state’?
•
Various common vector notations:
– Just a name, an abstraction that refers to
something physical
r
– Or alternatively as a vector in an M-dimensional
complex-valued vector space
•
Just as calculus provides the mathematical basis
for Classical Mechanics, the mathematical basis
for QM is linear algebra
– Vectors, matrices, eigenvalues, rotations, etc… are
key concepts
r
r
!
– Unit vectors are predefined in physical
r r
terms
r1 (t) = r " e1
– Components are projections onto unit
!
r r
r r
vectors
e !e = 0 e !e =1
1
2
1
– Unit vectors are orthonormal
1
!
3. Column vector:
– Unit vectors are implied
– We will refer to this abstract vector space as
`Hilbert Space’ or `state space’
– Any vector in this space corresponds to a possible
quantum-mechanical state. The number of such
quantum states is uncountable infinity
r
2. Unit vectors: r (t) = r1 (t)e1 + r2 (t)e2 + r3 (t)e3
Consider a system with M distinct available
states
– The 2M real numbers can be viewed as a vector in
an 2M-dimensional real-valued vector space
r
r (t)
1. Vector notation:
" r1 (t) %
r
$
'
r (t) = $ r2 (t)'
$
'
# r3 (t)&
`Dirac notation’:
–
Just new symbols for
! same concepts `ket’
r
r (t ) # ! (t ) , " (t ) , % (t ) ,...
`bra’
rT
r (t ) # ! (t ) , " (t ) , % (t ) ,...
r
e j # j , n , an , r , p , n, m , En , En , m ,...
r
r
r
r
r t ) = r1 (t )e1 + r2 (t )e2 + r3 (t )e3 # ! (t ) = c1 (t ) 1 + c2 (t ) 2 + c3 (t ) 3
r r r r
a $ b = aT b = a b
`inner product’
r r
rj (t ) = e j $ r (t ) # c j (t ) = j ! (t )
Added catch since QM vectors are complex
• Norm of a vector:
– Transpose operation replaced by ‘Hermitian
conjugation’ or ‘dagger’ operation
ba = ab
"
– a.k.a. magnitude, length
r r
r = r !r
r r
= rTr
! =
!!
– ‘†’ is transpose plus complex conjugation
!
r
rT
r T = (r ) " # = ( #
)
†
• To compute the norm in terms of the
components along a set of orthogonal unit
vectors:
• Projectors and Closure relations:
r
r
r
r r
! r = r1e1 + r2e2 + K + rM eM
r r r r r r
r r r
= e1 (e1 " r ) + e2 (e2 " r ) + K + eM (eM " r )
– Insert the identity
) ) = ) 1)
(M
%
= ) && ! j j ## )
' j =1
$
" = c1 1 + c1 1 + K + c M M
!
= 1 1" + 2 2 " +K+ M M "
M
=! ) j j)
= (1 1 + 2 2 +K+ M M )"
– This proves the `closure relation’:
!
j =1
M
"j
j=1
The summation is over a complete set of unit vectors that
spans any Hilbert sub-space is equal to the identity
operator in that sub-space
!
•The entire Hilbert space is a trivial sub-space
M
M
j =1
j =1
= ! c "j c j = ! c j
j =1
Old notation:
2
r r
r " r = r12 + r22 + K
= ! rj2
j
Avoid being confused by implied
meanings of various symbols
Summary
• There are ‘ket’s and ‘bra’s:
• To avoid confusion, keep in mind that ‘| !’
indicates a Hilbert-space vector, the ‘"’ in |" !’
is just a label
– We could call it anything
•
|" ! , |# ! , |$ ! , |3 ! , |Alice !
– We just need to clearly define our labels
• “let |"(t) ! be the state of our system at time t.”
• “let |x ! be the state in which the particle lies at
position x.”
– Here x is a placeholder which could take on any
numerical value. I.e. defining the state |x ! as above
actually defines an infinite set of vectors, one for
each point on the real axis.
– This is exactly how the symbol ‘x ’ is used when you
say `f(x) = cos(x)’
• ‘let |j ! be the state in which our system is in the jth
quantized energy level.
– Here j is a placeholder for an arbitrary integer
– ket: |" !
• A ket is a vector in an M dimensional Hilbert
space, where M is the number of distinct
physical states of a system
– bra: %"|
• A bra is a transposed, conjugated ket
• Put a bra and a ket together to get a cnumber
– %"|#! := a c-number
• c-number := complex number
• Unit vectors:
– An M dimensional Hilbert space is spanned by
M orthonormal unit vectors
– {|j!}={|1!, |2!, |3!, …, |M! } ( { } = ‘the set of’ )
– %j|k! = &jk
( &jk is ‘Kronecker delta function’)
» 1 if j=k
» 0 else
– Closure relation:
M
"j
j=1
!
j =1