IV.
Matrix Mechanics
We now turn to the a pragmatic aspect of QM: given a particular
problem, how can we translate the Dirac notation into a form that
might be interpretable by a computer? As hinted at previously, we do
this by mapping Dirac notation onto a
complex vector space
. The
operations in Hilbert space then reduce to linear algebra that can
easily be done on a computer. This formalism is completely
equivalent to the Dirac notation we’ve already covered; in different
contexts, one will prove more useful than the other.
1.
States can be represented by vectors
First, we will begin with an arbitrary complete orthonormal basis of
{
}
i
φ
.
states
Then, we know that we can write any other state as:
ψ
=
c
1
φ
1
+
c
2
φ
2
+
c
3
φ
3
=
+
...
ƒ
c
i
i
φ
i
How are these coefficient determined? Here, we follow a common
trick and take the inner product with the
j
th
state:
)
(
ƒ
(
)
)
(
φ
j
i
i
i
Since the Kronecker delta is only nonzero when i=j, the sum
collapses to one term:
ƒ
ƒ
j
ψ
φ
i
φ
i
j
φ
φ
δ
i
ij
=
=
=
c
c
c
i
i
j
ψ
φ
=
c
j
The simple conclusion of these equations is that
knowing the
coefficients is equivalent to knowing the wavefunction
. If we know
ψ
, we can determine the coefficients through the second relation.
ψ
cients, we can reconstruct
by
Vice versa, If we know the coeffi
performing the sum
ƒ
i
c
i
i
φ
. Thus, if we fix this arbitrary basis, we
can throw away all the basis state and just keep track of the
coefficients
of the ket state:
ψ
’
÷
÷
÷
÷
÷
◊
≈
Δ
Δ
Δ
Δ
Δ
«
c
1
c
2
→
c
3
...
φ
In harmony with the intuitive arguments made previously, here we
associate the ket states with column vectors. Notice the small
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subscript “
φ
”, which reminds us that this vector of coefficients
ψ
represents
i
{
}
i
φ
n the
basis. If we were really careful, we would
keep this subscript at all times; however, in practice we will typically
know what basis we are working in, and the subscript will be dropped.
ψ
ng bra state
How to we represent the correspondi
as a vector?
Well, we know that
†
=
(
)
†
ψ
=
≈
Δ
«
’
ƒ
ƒ
ψ
÷
◊
i
φ
i
φ
*
=
c
c
.
i
i
i
i
Now, as noted before, we expect to associate bra states with row
vectors, and the above relation shows us that the elements of this
row vector should be the
complex conjugates
of the column vector:
*
*
*
ψ
→
(
c
1
c
2
c
3
...
)
φ
Noting that bra states and ket states were defined to be Hermitian
conjugates of one another, we see that
Hermitian conjugation in state
space corresponds to taking the complex conjugate transpose of the
coefficient vector.
Now, the vector notation is totally equivalent to Dirac notation; thus,
anything we compute in one representation should be exactly the
same if computed in the other. As one illustration of this point, it is
useful to check that this association of states with vectors preserves
the inner product:
≈
’
=
≈
Δ
«
’
÷
÷
◊
j
φ
φ
φ
j
i
ij
ij
i
ƒ
i
φ
ƒ
ƒ
ƒ
ƒ
'
ψ
ψ
*
'
*
'
*
'
*
'
c
c
i
i
δ
j
ij
Δ
Δ
«
÷
◊
=
=
=
c
i
c
c
c
i
c
c
i
j
j
i
j
≈
’
c
1
c
2
Δ
Δ
Δ
Δ
Δ
«
÷
÷
÷
÷
÷
(
c
...
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 Spring '04
 RobertField
 Linear Algebra, basis, Hilbert space

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