The Heisenberg Uncertainty Principle

Chemistry 460
Spring 2015
Dr. Jean M. Standard
March 2, 2015
The Heisenberg Uncertainty Principle
A policeman pulls Werner Heisenberg over on the Autobahn for speeding.
Policeman: Sir, do you know how fast you were going?
Heisenberg: No, but I know exactly where I am.
I. Definition and Examples
The most general form of the Heisenberg Uncertainty Principle is
ΔA ΔB ≥
1
2
[ Aˆ , Bˆ ]
(1)
,
where Aˆ and Bˆ are Hermitian operators. The uncertainty ΔA in the physical observable A associated with the
operator Aˆ is defined by the relation
€
€
€
ΔA€=
€
[A
2
− A
2 1/ 2
]
(2)
.
The Heisenberg Uncertainty Principle can be evaluated for specific operators. As an example, consider Aˆ = pˆ x and
Bˆ = xˆ . The commutator is [ pˆ x , xˆ ]€= − i ! . Then, the Heisenberg Uncertainty Principle (HUP) takes the form
€
Δpx Δx ≥
€
1
2
[ pˆ x , xˆ ]
.
€
(3)
The commutator is
€
[ pˆ x , xˆ ]
= −i! .
(4)
Then, the uncertainty principle becomes
€
Δpx Δx ≥
=
=
=
=
Δpx Δx ≥
€
1
[ pˆ x , xˆ ]
2
1
ψ −i! ψ
2
1
−i! ψ ψ
2
1/ 2
1
(−i !)* (−i !)
2
1/ 2
1
{
(i !) (−i !)}
2
!
.
2
{
}
(5)
2
II. Comparison Between Quantum Uncertainty and Standard Deviation
The uncertainty of an observable A in quantum mechanics is defined as
$
Δ A = & A2 − A
%
')1/ 2
,
(
2
(6)
where the expectation value is
€
A
∫ ψ * Aˆ ψ dτ
∫ ψ * ψ dτ
=
.
(7)
In statistics, the standard deviation σ is defined as the "root mean squared" deviation, and can be written
€
n
%
( 1/2
2
1
σ = '
Ai − A ) *
,
(
'& n i=1
*)
∑
where n is the sample size and
(8)
A denotes the average,
€
A
€
=
1
n
n
∑
Ai .
(9)
i=1
Note that in the definition of the standard deviation given in Eq. (8), a large sample size has been assumed.
Otherwise, the factor 1/n should be 1/(n–1).
€
Eq. (8) can be rearranged to give a form very similar to the definition of the quantum mechanical uncertainty.
Starting with the uncertainty given in Eq. (8), we can expand out the square,
n
∑
( Ai − A
)
2
n
=
i=1
∑ (A
2
i
− 2 Ai A + A
i=1
n
=
=
∑
)
2
n
Ai2 − 2
∑
i=1
n
i=1
∑
Ai2 − 2 A
n
Ai A +
∑
A
(10)
i=1
n
i=1
2
n
∑
Ai + A
2
∑1 .
i=1
i=1
Using definitions of the averages,
€
A
€
=
1
n
n
∑ Ai
i=1
and
A2
=
1
n
n
∑ Ai2 ,
i=1
(11)
3
we can solve and obtain
n
n
∑
Ai = n A
∑ Ai2
and
i=1
= n A2 .
(12)
i=1
Substituting Eq. (12) into Eq. (10),
€
n
∑ (A
i
− A
)
2
= n A2
− 2 A n A
= n A2
− 2n A
+ A
2
n
i=1
∑ (A
i
− A
)
2
= n
i=1
(
A2 − A
+ n A
2
2
= n A2 − n A
n
2
2
(13)
).
The relation obtained in Eq. (13) can be inserted into Eq. (8), which gives the standard deviation,
€
n
%
( 1/2
2
1
σ = '
A
−
A
( i
) **
'& n i=1
)
1/2
% 1
2 (
= '
n A2 − A
*)
& n
∑
(
σ =
(
A2 − A
)
2
1/ 2
)
(14)
.
Comparing the statistical standard deviation in Eq. (14) to the quantum mechanical uncertainty given in Eq. (6), we
see that they are identical.
€
4
III. Why the Commutator Plays a Role in the Uncertainty Principle
It can be shown that if two operators Aˆ and Bˆ commute, then they can have the same set of eigenfunctions (this is
referred to as simultaneous eigenfunctions). In other words, suppose ψ i corresponds to the set of eigenfunctions of
the operator Aˆ , such that
€
€
(15)
Aˆ ψ = a €
ψ .
i
i
i
€
Then, if Aˆ and Bˆ commute such that Aˆ , Bˆ
€
Bˆ ,
[
€
€
]
= 0 , the functions ψ i also are a set of eigenfunctions of the operator
Bˆ ψ i = b€
i ψi .
€
(16)
€
If the operators Aˆ and Bˆ commute, then
€ from the Heisenberg Uncertainty Principle the observable properties A and
B may be measured exactly,
€
ΔA ΔB ≥ 0 .
€
(17)
This result is also in accord with the Measurement Postulates of quantum mechanics. The first Measurement
Postulate states that if a system is in a state ψ that is an eigenfunction of the operator corresponding to the property,
€
then measurement yields the eigenvalue exactly. If we wish to measure two properties simultaneously and exactly,
the only way we may do this is if the state ψ is an eigenfucntion for both operators.
€
€
5
APPENDIX 1. The Momentum Representation
We generally express the wavefunction of a system in terms of position; i.e., ψ ( x ) . However, it is possible and
sometimes desirable to convert to a representation in which the momentum is the variable upon which the
wavefunction depends. The position and momentum representations of the wavefunction are related by a Fourier
Transform,
€
φ ( px ) =
1
2π
∞
∫ ψ ( x) e
−ip x x / !
dx .
(1)
−∞
2
The momentum space probability density, φ ( px ) , is defined as the probability density for finding the particle with
a momentum value between p €and p + dp .
x
x
x
To get back to the position representation,
the inverse Fourier Transform is employed,
€
€
€
ψ( x) =
€
1
! 2π
∞
∫
−∞
φ ( px ) e +ip x x / ! dpx .
(2)
6
APPENDIX 2. Derivation of the Heisenberg Uncertainty Principle
There are five steps involved in the derivation of the Heisenberg Uncertainty Principle:
1.
Proof of the Schwarz Inequality.
2.
Proof of
αβ
3.
€Proof of α β
4.
€Proof of 1
4
2
*
≤ α2
=
[ αˆ , βˆ ]
β 2 , where αˆ and βˆ are general Hermitian operators.
€
β α , where
αˆ €and βˆ are the same general Hermitian operators as in step 2.
2
€
= b 2€, where α β
α2
€
€
= a + i b . Also, we use step 2 to show that
β2
1
4
≥
2
[ αˆ , βˆ ]
.
ˆ
ˆ
ˆ
Use of αˆ = A − A , and€ β = B − B to generate the Uncertainty Principle.
5.
€
€
1. Proof of Schwarz Inequality
We wish to prove the Schwarz Inequality,
ψ φ
2
≤
ψ ψ
φ φ ,
where ψ and φ are arbitrary functions. To begin the proof, we define
€
€
f (λ ) =
φ − λψ φ − λψ ,
(1)
€
where λ is a parameter (that may be complex). Note that for some function f(x) (which could be complex), we can
write f ( x ) = a ( x ) + i b( x ) in order
€ to split f(x) into its real and imaginary parts. The functions a( x ) and b( x )
are then real functions. The absolute square of the function f(x) is defined as
€
f ( x)
€
f ( x)
2
2
= f *( x ) f ( x)
€
=
{ a( x)
− i b( x )} { a( x ) + i b( x )}
=
{ a( x)}
2
2
+ {b( x )} .
Since a( x ) and b( x ) are real, we know for all x that
€
€
€
€
{a( x )} 2
≥ 0 and
{b( x )} 2
≥ 0.
€
7
Therefore,
f ( x)
2
≥ 0.
Because of this, we also have for any function
€
2
∫
f ( x ) dx ≥ 0.
Using this relation and the expression for f ( λ ) from Eq. (1) above,
€
f (λ ) =
€
=
=
φ − λψ φ − λψ
∫ (φ − λψ)*(φ − λψ) dτ
∫ (φ − λψ) 2 dτ ≥ 0
(2)
f ( λ ) ≥ 0.
The parameter λ is arbitrary. It can be chosen to be any number. To get the Schwarz Inequality, select
€
λ =
ψ φ
.
ψ ψ
λ* =
φ ψ
.
ψ ψ
€
Then,
€
The expression for f ( λ ) , Eq. (1), can be expanded to yield
f (λ ) =
€
=
€
φ − λψ φ − λψ
φ φ
− λ* ψ φ
− λ φ ψ
+ λ*λ ψ ψ .
Substituting the choices for λ and λ* gives
€
f (λ ) =
φ φ
− λ* ψ φ
€
€
φ φ
−
=
φ ψ
ψ ψ
+ λ*λ ψ ψ
− λ φ ψ
ψ φ
ψ φ
ψ ψ
−
φ ψ
+
φ ψ
ψ ψ
ψ φ
ψ ψ
ψ ψ .
The last two terms cancel , which leads to
€
f (λ ) =
φ φ
−
φ ψ
ψ ψ
ψ φ .
Since from Eq. (2), we have that f ( λ ) ≥ 0 , Eq. (3) can be rewritten as
€
φ φ
€
Rearranging,
€
−
φ ψ
ψ ψ
ψ φ
≥ 0.
(3)
8
φ φ
Since
φ ψ
ψ φ
=
*
ψ ψ
ψ φ .
φ ψ
≥
, this equation becomes the Schwarz Inequality,
€
φ φ ψ ψ ≥ φ ψ ψ φ ,
€
φ φ
ψ ψ
φ φ
≥
ψ ψ
*
ψ φ
≥
2
ψ φ
ψ φ ,
.
Alternately, this equation can be written as
€
2. Proof of
αβ
2
≤ α2
ψ φ
2
≤
ψ ψ .
φ φ
€
β 2 , where αˆ and βˆ are general Hermitian operators.
To show this relation, we must use the definition of the complex conjugate of an integral,
€
€
€
*
f Oˆ g
= Oˆ g f .
Starting with the left side of the relation that we wish to prove, we have
€
αβ
2
=
=
2
ψ αˆ βˆ ψ
ψ αˆ βˆ ψ
= αˆ βˆψ ψ
*
ψ αˆ βˆ ψ
(4)
ψ αˆ βˆ ψ .
Next, we use the definition of a Hermitian operator. If Oˆ is Hermitian, then
€
f Oˆ g
= Oˆ f
g .
€
Eq. (4) can therefore be written as
€
€
αβ
2
= αˆ βˆψ ψ
ψ αˆ βˆ ψ
= βˆψ αˆ ψ
αˆ ψ βˆψ .
(5)
9
*
f Oˆ g
Once again, the complex conjugate of an integral is utilized,
= Oˆ g f . Substituting this into Eq. (5)
yields
α β€
2
= βˆψ αˆ ψ
αβ
*
αˆ ψ βˆψ
=
2
αˆ ψ βˆψ
2
αˆ ψ βˆψ
=
αˆ ψ βˆψ
(6)
.
Now the Schwarz Inequality can be used, with the replacements
€
ψ → αˆψ ,
φ → βˆ φ .
Substituting,
αβ
2
€
=
2
αˆ ψ βˆψ
≤
βˆ ψ βˆψ
αˆ ψ αˆψ .
If the operators αˆ and βˆ are Hermitian,
€
€
βˆ ψ βˆψ
€
= ψ βˆ 2 ψ
αˆ ψ αˆψ
=
ψ αˆ 2 ψ
=
=
β2
α2 .
Finally, these relations can be used in Eq. (7) to yield the desired result,
€
αβ
or
3. Proof of α β
*
αβ
2
2
≤
βˆ ψ βˆψ
≤
β2
αˆ ψ αˆψ
α2 .
€
β α , where αˆ and βˆ are the same general Hermitian operators as in step 2.
=
Start with the left side,
€
€
€
αβ
*
=
ψ αˆ βˆ ψ
*
.
Use the definition of the complex conjugate of an integral,
€
f Oˆ g
*
= Oˆ g f ,
to get
€
αβ
*
=
ψ αˆ βˆ ψ
*
= αˆ βˆ ψ ψ .
Using the Hermitian character of αˆ leads to
€
€
(7)
10
*
αβ
= αˆ βˆ ψ ψ
=
βˆ ψ αˆψ .
=
βˆ ψ αˆψ
=
ψ βˆ αˆ ψ .
Using the Hermitian character of βˆ leads to
€
αβ
*
€
Therefore, this shows that
€
αβ
or
1
4. Proof of
4
[
αˆ , βˆ
]
2
αβ
€
= b 2 , where α β
*
*
=
ψ βˆ αˆ ψ
=
βα .
= a + i b.
Since an expectation value is just a number (which might be complex), we define
€
€
α β = a + i b,
where a and b are real numbers. From this definition and Part 3 of the Proof,
€
βα
αβ
=
*
= ( a + i b)
= a − i b.
βα
Subtracting
βα
= a − i b from α β
€
αβ
€
*
= a + i b gives
−
= a + i b − ( a − i b)
βα
= 2i b.
€
This expression can be rewritten in terms of an expectation value of the commutator,
€
βα
= 2i b
α β − βα
= 2i b
αβ
−
[ αˆ , βˆ ]
= 2i b.
Using the properties of integrals, we also have
€
Then,
€
[ αˆ , βˆ ]
*
= − 2i b.
11
2
[ αˆ , βˆ ]
*
[ αˆ , βˆ ]
=
[ αˆ , βˆ ]
= (−2 i b)( 2i b)
2
[ αˆ , βˆ ]
= 4b 2 .
Solving for b2,
€
b2 =
2
[ αˆ , βˆ ]
1
4
(8)
.
In addition, from Part 2 of the Proof,
€
α2
β2
2
αβ
≥
.
The right side of this equation can be written as
€
αβ
2
=
αβ
*
αβ
= ( a − ib)( a + ib)
αβ
2
= a
2
(9)
2
+ b .
From Part 2 of the Proof,
€
α2
2
β2
≥
β2
≥ a 2 + b2 .
αβ
.
From Eq. (9) we can substitute for the right side,
€
α2
2
2
2
2
2
2
Since a ≥ 0 and b ≥ 0 , it must be true that a + b ≥ b . Using the expression for b from Eq. (8),
€
€
€
€
α2
β2
€
or α 2
2
β
≥ a 2 + b2 ≥ b2
≥
1
4
[
αˆ , βˆ
]
2
€
.
(10)
12
ˆ
5. Use of αˆ = A − A , and βˆ = Bˆ − B to generate the Uncertainty Principle.
ˆ
ˆ
ˆ
Let αˆ = A − A and β = B − B . Then,
€
€
α2
€
(A −
=
A 2 − 2A A + A
€
α2
)
2
=
A
=
A2 − 2 A
=
A2 −
A
2
2
2
2
+ A
.
Therefore, using the definition of the quantum mechanical uncertainty,
€
2
α2
= ( ΔA) .
2
2
Similarly, carrying out the same expansion for β , we can show that β
€
ˆ
ˆ
α
The commutator of and β is given by
€
€
€
[ αˆ , βˆ ] = [ Aˆ − A , Bˆ − B ]
= [ Aˆ , Bˆ ] − [ Aˆ , B ] − [
€
2
= ( ΔB) .
A , Bˆ +
] [
A , B ].
(11)
Since A and B are just constants, they commute with any other operators. Therefore, the last three terms in Eq.
(11) are zero. The commutator
becomes
€
€
[ αˆ , βˆ ] = [ Aˆ , Bˆ ] .
€
From Part 4 of the Proof, we have
€
α2
β2
≥
1
4
[ αˆ , βˆ ]
2
.
Substituting the definitions of αˆ and βˆ , this relation becomes
€
€
( ΔA) 2 ( ΔB) 2
€
≥
1
4
[ Aˆ , Bˆ ]
2
.
Taking the square root of both sides gives the Heisenberg Uncertainty Principle,
€
( ΔA)( ΔB)
€
≥
1
2
[ Aˆ , Bˆ ]
.