Heisenberg's Uncertainty Principle - Wichita State Universitymelin/research/classes/2016-08-fall/phys-714/phys714...1 Vector Space 2 Banach Space 3 Hilbert Space 4 Self-adjoint Operators

Heisenberg’s Uncertainty Principle

Jaron P. Melin

Wichita State University

2016-10-20

Jaron P. Melin (Wichita State University) Heisenberg’s Uncertainty Principle 2016-10-20 1 / 26

1 Vector Space

2 Banach Space

3 Hilbert Space

4 Self-adjoint Operators

5 Heisenberg’s Uncertainty Principle

6 Position- and Momentum-Operators


Definition

Let V be a set where addition and scalar multiplication are defined, i.e., ifx , y ∈ V and α ∈ C, then x + y ∈ V and αx ∈ V . The set together withthese algebraic operations form a complex vector space if the followingaxioms are satisfied:

1 x + y = y + x for any x , y ∈ V ;

2 (x + y) + z = x + (y + z) for any x , y , z ∈ V ;

3 There exists some element 0 ∈ V such that x + 0 = x for any x ∈ V ;

4 For every x ∈ V , there exists some element −x ∈ V such thatx + (−x) = 0;

5 α(x + y) = αx + αy for any α ∈ C and x , y ∈ V ;

6 (α + β)x = αx + βx for any α, β ∈ C and x ∈ V ;

7 (αβ)x = α(βx) for any α, β ∈ C and x ∈ V ;

8 1x = x for any x ∈ V .


Banach Space

Definition

Consider any complex vector space X . Any function || · || : X → R is anorm if it has the following properties:

1 (positivity) ||x || > 0 for any x 6= 0, and ||0|| = 0;

2 (sub-additivity) ||x + y || ≤ ||x ||+ ||y || for any x , y ∈ X ;

3 (homogeneity) ||ax || = |a| ||x || for any x ∈ X and a ∈ C.

From this norm, we may define a metric

d : X × X → R : (x , y) 7→ ||x − y ||.


Definition

An open ball of radius r centered at some fixed y ∈ X is denoted by

Br (y) = {x ∈ X : ||x − y || < r}.

The collection of open balls {Br (x) ⊂ X : x ∈ V , 0 < r ∈ Q} may act as abase for the topology on X . So then, any open or closed set in X may beconstructed from this base. Combined with this topology, X may be calleda topological vector space. From here, the notions of convergence andcontinuity may ensue.


Definition

Any sequence {xn ∈ X : n ∈ N} converges to some x ∈ X if

limn→∞

||xn − x || = 0.

Definition

Any sequence {xn ∈ X : n ∈ N} is a Cauchy sequence if

limm,n→∞

||xm − xn|| = 0.

As a consequence, every convergent sequence is indeed a Cauchysequence, but a Cauchy sequence is not necessarily a convergent sequence.


Definition

A normed vector space X is complete if every Cauchy sequence converges.

Definition

A complete normed vector space is a Banach space.


Hilbert Space

Definition

Consider any complex topological vector space X . Any form〈·, ·〉 : X × X → C is an inner product on X if has the followingproperties:

1 (sesquilinearity) for any x , y , z ∈ X and a, b ∈ C,

〈ax + by , z〉 = a〈x , z〉+ b〈y , z〉〈x , ay + bz〉 = a〈x , y〉+ b〈x , z〉;

2 (skew symmetry) 〈x , y〉 = 〈y , x〉 for any x , y ∈ X ;

3 (positivity) 〈x , x〉 > 0 for any x 6= 0.

Any inner product yields a norm such that for x ∈ X ,

||x || =√〈x , x〉. (1)


Theorem (Schwarz Inequality)

For any inner product,|〈x , y〉| ≤ ||x || ||y ||,

where the norm is defined by (1). Equality holds either when x = ky fork ∈ C or when y = 0.

Definition

A topological vector space H with an inner product 〈·, ·〉 which is completewith respect to its induced norm is a Hilbert space. In other words, aHilbert space is just a Banach space with an inner product.

If some topological vector space has an inner product whose induced normis not complete, then the space can be completed by virtue of thecontinuity of the inner product in its factors. The completion then is aHilbert space.


Example

The space

L2(R) =

{u : R→ C

∣∣∣∣ ∫R|u(x)|2 dx <∞

}of square-integrable functions with inner product

〈f , g〉 =

∫Rf (x)g(x) dx

is a Hilbert space.


Self-adjoint Operators

Definition

Any function A : X → Y of complex topological vector spaces is a linearoperator if

A(ax + by) = aAx + bAy

for any a, b ∈ C and x , y ∈ X .

Definition

Any linear operator A : X → Y of Banach spaces is bounded if thereexists some constant c ∈ R such that for any x ∈ X ,

||Ax || ≤ c ||x ||.


Definition

Let H be any complex Hilbert space, D any dense subspace of H, and Aany linear operator with dom(A) = D. The linear operator A isself-adjoint if for any u, v ∈ H,

〈Au, v〉 = 〈u,Av〉.

According to Hellinger and Toeplitz, if a linear operator A is definedeverywhere on a Hilbert space H so that dom(A) = H and A is alsoself-adjoint, then A is necessarily bounded.


Heisenberg’s Uncertainty Principle

In quantum mechanics, the following associations are made:

physical system←→ complex Hilbert spaceH

state of the system←→ unit vector inH

observable←→ self-adjoint operator.

Example

Some examples of observables are position, angular momentum, spin, andthe Hamiltonian for the total energy of a particle of mass m in a realpotential field V such that Hψ = −~2

2m ∇2ψ + Vψ. Notice that this

Hamiltonian is a differential operator; differential operators are oftenunbounded.


Definition

The expected value of an observable A in a state u ∈ ∂B1(0) ∩ dom(A)is defined by

EV(A, u) := 〈u,Au〉.

Example

(finite case) If dim(H) = n ∈ N, u = (u1, . . . , un), Au = (p1, . . . , pn)where

∑ni=1 pi = 1, and 〈u, v〉 = u · v for u, v ∈ dom(A), then

EV(A, u) = 〈u,Au〉 = u · Au =n∑

i=1

uipi =

∑ni=1 uipi∑ni=1 pi

,

which can be viewed as a kind of weighted average.


Example

The expected value of rolling a six-sided die is given by

EV(A, u) =6∑

i=1

ipi =6∑

i=1

i

(1

6

)=

1

6

6∑i=1

i =1

6

6(6− 1)

2=

5

2= 3.5

By using the term “expected value”, it implies that there is someuncertainty in the measurement of the observable in some state.

Definition

The uncertainty in the measurement of an observable A in some stateu ∈ ∂B1(0) ∩ dom(A) is defined by

∆(A, u) :=√

EV((A− aI )2, u),

where a = EV(A, u).


Note that

∆(A, u)2 = EV((A− aI )2, u)

= 〈u, (A− aI )2u〉= 〈u,A2u〉 − 2a〈u,Au〉+ a2〈u, u〉= 〈Au,Au〉 − 2a(a) + a2

= ||Au||2 − a2. (2)

This means that ∆(A, u) = 0 only if Au = au; in other words, absolutecertainty is obtained only if u is an eigenstate of A. Furthermore, if A,Bare two distinct observables, then ∆(A, u) = 0 and ∆(B, u) = 0 if andonly if Au = au and Bu = bu; that is, the observables A,B can bemeasured with absolute certainty if and only if u is an eigenvector of bothA and B at the same time.


If two distinct observables A,B satisfy additionally the Heisenbergcommutation-relation

[A,B] = AB − BA = i idD , (3)

where D := dom(A) ∩ dom(B) ∩ dom(AB) ∩ dom(BA), then A and Bwould have no common eigenvectors between them. This would mean thatthese observables cannot both be measured with absolute certainty.

Theorem (Heisenberg’s Uncertainty Principle)

Suppose that A,B are two distinct observables which satisfy theHeisenberg commutation-relation (3). It follows that for every stateu ∈ ∂B1(0) ∩ D,

∆(A, u)∆(B, u) ≥ 1

2.


Proof

Note that by (3),

〈Bu,Au〉 − 〈Au,Bu〉 = 〈ABu, u〉 − 〈BAu, u〉= 〈(AB − BA)u, u〉= 〈[A,B]u, u〉= 〈iu, u〉= i〈u, u〉= i||u||2. (4)

for any u ∈ D. Now, take any u ∈ ∂B1(0) ∩ D and t ∈ R, and notice thatby Schwarz’s inequality,

|〈u,Au + itBu〉|2 ≤ ||u||2||Au + itBu||2 = ||Au + itBu||2. (5)

It is desired to simplify this inequality.


Let a := EV(A, u) and b := EV(B, u). See that

|〈u,Au + itBu〉|2 = |(〈u,Au〉+ 〈u, itBu〉)|2

= |(〈u,Au〉 − it〈u,Bu〉)|2

= |(a− ibt)|2

= (a)2 + (−bt)2

= a2 + b2t2. (6)


Furthermore, see that by (4),

||Au + itBu||2 = 〈Au + itBu,Au + itBu〉= 〈Au,Au + itBu〉+ 〈itBu,Au + itBu〉= 〈Au,Au〉+ 〈Au, itBu〉+ 〈itBu,Au〉+ 〈itBu, itBu〉= ||Au||2 − it〈Au,Bu〉+ it〈Bu,Au〉 − i2t2〈Bu,Bu〉= ||Au||2 + it(〈Bu,Au〉 − 〈Au,Bu〉) + ||Bu||2t2

= ||Au||2 + it(i||u||2) + ||Bu||2t2

= ||Au||2 − t + ||Bu||2t2. (7)


By combining (6) and (7) into (5) and using (2), obtain

|〈u,Au + itBu〉|2 ≤ ||Au + itBu||2

a2 + b2t2 ≤ ||Au||2 − t + ||Bu||2t2

0 ≤ (||Au||2 − a2)− t + (||Bu||2 − b2)t2

0 ≤ ∆(A, u)2 − t + ∆(B, u)2t2

On the right-hand side of this inequality, there is a quadratic function withrespect to t. Since this quadratic function is non-negative for any t ∈ R,then its discriminant non-positive so that

(−1)2 − 4∆(B, u)2∆(A, u)2 ≤ 0

∆(B, u)2∆(A, u)2 ≥ 1

4

∆(A, u)∆(B, u) ≥ 1

2.

�


Example: Position- and Momentum-Operators

Comment

What pairs of operators actually satisfy (3)? According to Weilandt, nobounded operators do. So then, A and B must be unbounded operators.

Let H = L2(R). Consider the linear operators A and B given by

Au(x) = xu(x), dom(A) = {u ∈ L2(R) : xu(x) ∈ L2(R)}Bu(x) = −iu′(x), dom(B) = {u ∈ L2(R) : u′ ∈ L2(R)} .

Here, A is the position-operator and B is the momentum-operator.



Notice that for any u ∈ dom(A),

〈u,Au〉 =

∫Ru(x)Au(x) dx

=

∫Ru(x) · xu(x) dx

=

∫Rxu(x) · u(x) dx

=

∫RAu(x)u(x) dx

= 〈Au, u〉.



Notice also from integration-by-parts that for any u ∈ dom(B),

〈u,Bu〉 =

∫Ru(x)Bu(x) dx

=

∫Ru(x) · −iu′(x) dx

=

∫Ru′(x) · iu(x) dx

=

∫R−iu′(x) · u(x) dx

=

∫RBu(x)u(x) dx

= 〈Bu, u〉.

So then, A and B are self-adjoint operators.



Furthermore, for any u ∈ D, where D is defined as in (3),

[A,B]u = (AB − BA)u

= ABu − BAu

= A(−iu′)− B(xu)

= x(−iu′)− (−i)(xu)′

= −ixu′ + i(u + xu′)

= −ixu′ + iu + ixu′

= iu

Hence, A and B satisfy the Heisenberg commutation-relation (3).Therefore, by Heisenberg’s Uncertainty Principle, ∆(A, u)∆(B, u) ≥ 1

2 .According to von Neumann, this pair of observables is the only pair uptomultiplicity and unitary equivalence which satsifies (3).


References

1 Lax, Peter. Functional Analysis. New York: Wiley-Interscience,2002. Pages 8, 36, 38, 52-53, 160, 377-378, 455-457.

2 Leon, Steven. Linear Algebra With Applications. 7th edition.Upper Saddle River, NJ: Pearson Prentice Hall, 2006. Pages 118-119.

3 Sieradski, Allan J. An Introduction to Topology and Homotopy.PWS Publishers, 1996. Page 153.


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Heisenberg's Uncertainty Principle - Wichita State Universitymelin/research/classes/2016-08-fall/phys-714/phys714...1 Vector Space 2 Banach Space 3 Hilbert Space 4 Self-adjoint Operators