lec10 infinite dimension vector spaces

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Lecture 10

Infinite Dimensional Vector Spaces

October 6, 2010

Lecture 10


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Infinite Dimensional Space

Consider the set of continuous, single-variable functions,

f(x), on the intervalL/2

x

L/2.

If we specify these functions by giving there values at nequally-spaced, discreet points, xi, they form vectors, |f,in an n-dimensional vector space.

We can choose the orthonormal basis vectors to those

specifying each of the discreet point.

|i =

00...

0

10...

0

with i|j = ij andni=1

|ii| = I

The components of the vector |f are then the values ofthe function at the discreet points

|f =ni=1

i|f|i with i|f = f(xi)

Lecture 10 1


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Operators in Infinite Dimension Space

An operator, , in an infinite dimensional space transformsa vector, |f, in the vector space to another vector, |f, inthe space.

|f = |gIn the x basis

x||f = x|g = g(x)

is represented by an infinite dimension matrix.

xx = x||x

x

|g

=

x

|

|x

x

|f

dx

Formally, we can write

f(x) = g(x)

What this really means though is

x||xx|fdx = g(x)

Lecture 10 5


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The Position Operator

Consider the operator, x, that when operating on a

function, f(x), has the effect of multiplying the functionby x.

x|x|f = xx|f = xf(x)

x|x|xx|fdx = xf(x)

x|x|x = x(x x)

The eigenvectors of x are the basis vectors, |x.

x|x|x0 = x0(x x0) = x0x|x0

As weve noted before, these eigenvectors have a delta

function rather than unity normalization.

Lecture 10 6


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The Derivative Operator

Another natural operator to consider is the operation of

taking a derivative of a function.

x|d|f = f(x)

x|d|f =

x|d|xx|fdx = ddx

f(x)

What are the matrix elements

x

|d

|x

?

Lecture 10 7


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The Derivative of the Delta Function

The matrix elements of D are given by the derivative of

the delta functionx|d|x = (x x) = d

dx((x x))

(x x)f(x)dx =

d

dx((x x)) f(x)dx

=

d

dx((x x)) f(x)dx

=

(x x)

d

dxf(x)

dx +

f(x)(x x)

=

(x x)

d

dxf(x)

dx =

d

dxf(x)

Lecture 10 8


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The k Operator

The operator d is not Hermitian.

dxx = (x x) = d

dx(x x) = (x x) d

dx

dxx = dxx = Dxx =

(x x) = ddx

(x x)

= (x x) ddx

= (x x) ddx

= dxx

Define the operator k = id

kxx = i

(x x

)

kxx = kxx = kxx = i(xx) = i(xx) = kxx

For an infinite dimension operator it is not sufficient that

x|k|x = x|k|x

We must also show that

g|k|f = f|k|g

Lecture 10 9


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Hermitivity of an Infinite Dimension

Operator

g|k|f = f|k|g

b

a

b

a

g|xx|K|xx|fdxdx = b

a

b

a

f|xx|K|xx|gdxdx

ib

a

g(x)

df(x)

dx

dx = i

ba

dg(x)

dx

f(x)dx

but the left-hand side is

ig(x)f(x)b

a+ i

ba

dg(x)

dx

f(x)dx

Hermitivity requires that f(x) or g(x) go to zero atthe limits a and b.

Lecture 10 10


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Eigenvectors of k

Let |k be an eignevector ofk with eigenvalue k.

k|k = k|k

Lets work in the x basis.

x|k|k = kx|k = kk(x)

x|k|xx|kdx = kk(x)

i ddx

k(x) = kk(x)

Solution: k(x) = Aeikx

Lecture 10 11


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Normalization

Now lets normalize |k.

k|k =

k|xx|kdx = |A|2

ei(kk)xdx

= |A|2 2(k k)

Choose: A =12

k|k = (k k)

As for the eigenvectors of x, these have delta function

normalization rather than unity normalization.

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Fourier Transforms

Lets find the vector

|f

in the k-basis.

f(k) = k|f =

k|xx|fdx

= 12

eikxf(x)dx

Similarly

f(x) = x|f =

x|kk|fdk

=1

2

eikxf(k)dk

f(k) and f(x) are the Fourier transforms of each other.

Lecture 10 13


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Canonical Commutation Relation

The operators x and k do not commute.

x|x|f = xf(x)

x|k|f = i df(x)dx

x|xk|f = ix df(x)dx

x|kx|f = ix ddx

xf(x)

x|[x,k]|f = x|[x,

k]|f x|[

k, x]|f

= ix df(x)dx

+ ixdf(x)

dx+ if = if = ix|I|f

[x, k] = iI

This means that there are no vectors that are eigenvectors

of both x and k. As you know and as well see, this leads toone of the most important and basic features of quantum

mechanics.

Lecture 10 14


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Reciprocity Between x and k

Representations

|xx| dx = I

|kk| dk = I

|f =

|xf(x) dx =

|kf(k) dk

x|x

= (x x

) k|k

= (k k

)

x|x|x = x(x x) x|k|x = i(x x)

k|x|k = i(k k) k|k|k = k(k k)

x|x|f = xf(x) x|k|f = if(x)

k|x|f = if(k) k|k|f = kf(k))

x|k =1

2 eikx

k|x =1

2 eikx

f(x) = x|f =

x|kf(k) dk = 12

eik

xf(k) d

Lecture 10 15


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The Momentum Operator

The momentum operator is simply the k

operator multiplied by h.

p = hk

x|p|f = ihf(x)

[x, p] = ihI

Lecture 10 16


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Correspondence with the Wavefunction

Formalism

| =

|xx| dx =

|x(x) dx

(x) gives the components of | in the |x basis.

| = |pp| dp = |p(p) dp(p) gives the components of | in the |p basis.

x

|p

are the matrix elements of the unitary

transformation from the p basis to the x basis.

(x) = x| =

x|p(p) dp = 12h

eipx/h(p) d

Lecture 10 17

lec10 infinite dimension vector spaces

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