The Heisenberg Uncertainty Principle

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Ἥβη Ζυγομαλάς
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1 Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig? Heiseberg: No, but I kow exactly where I am. I. Defiitio ad Examples The most geeral form of the Heiseberg Ucertaity Priciple is ΔA ΔB 1 A ˆ [, B ˆ ], (1) where A ˆ ad B ˆ are Hermitia operators. The ucertaity ΔA i the physical observable A associated with the operator A ˆ is defied by the relatio ΔA = [ A A ] 1/. () The Heiseberg Ucertaity Priciple ca be evaluated for specific operators. As a example, cosider A ˆ = p ˆ x ad B ˆ = x ˆ. The commutator is p ˆ x, x ˆ [ ] = i!. The, the Heiseberg Ucertaity Priciple (HUP) takes the form Δp x Δx 1 [ p ˆ x, x ˆ ]. (3) The commutator is [ p ˆ x, x ˆ ] = i!. (4) The, the ucertaity priciple becomes Δp x Δx 1 = 1 [ p ˆ x, x ˆ ] ψ i! ψ = 1 i! ψ ψ {( ) * ( i!)} 1/ = 1 i! {( )( i!)} 1/ = 1 i! (5) Δp x Δx!.

2 II. Compariso Betwee Quatum Ucertaity ad Stadard Deviatio The ucertaity of a observable A i quatum mechaics is defied as Δ A = $ & A A ' ) 1/, % ( (6) where the expectatio value is A = ψ * ˆ A ψ dτ ψ * ψ dτ. (7) I statistics, the stadard deviatio σ is defied as the "root mea squared" deviatio, ad ca be writte σ = % ' &' 1 ( 1/ ( A i A ) *, )* (8) where is the sample size ad A deotes the average, A = 1 A i. (9) Note that i the defiitio of the stadard deviatio give i Eq. (8), a large sample size has bee assumed. Otherwise, the factor 1/ should be 1/( 1). Eq. (8) ca be rearraged to give a form very similar to the defiitio of the quatum mechaical ucertaity. Startig with the ucertaity give i Eq. (8), we ca expad out the square, ( A i A ) = A i A i A + A = = A i ( ) A i A + A A i A A i + A 1. (10) Usig defiitios of the averages, A = 1 A i ad A = 1 A i, (11)

3 3 we ca solve ad obtai A i = A ad A i = A. (1) Substitutig Eq. (1) ito Eq. (10), ( A i A ) = A A A + A = A A + A = A A (13) ( A i A ) = A A ( ). The relatio obtaied i Eq. (13) ca be iserted ito Eq. (8), which gives the stadard deviatio, σ = = % ' &' % & ' 1 ( 1/ ( A i A ) * )* 1/ ( ) * ( ) 1 A A (14) ( ) 1/. σ = A A Comparig the statistical stadard deviatio i Eq. (14) to the quatum mechaical ucertaity give i Eq. (6), we see that they are idetical.

4 4 III. Why the Commutator Plays a Role i the Ucertaity Priciple It ca be show that if two operators A ˆ ad B ˆ commute, the they ca have the same set of eigefuctios (this is referred to as simultaeous eigefuctios). I other words, suppose ψ i correspods to the set of eigefuctios of the operator A ˆ, such that The, if A ˆ ad B ˆ commute such that B ˆ, ˆ A ψ i = a i ψ i. (15) A ˆ [, B ˆ ] = 0, the fuctios ψ i also are a set of eigefuctios of the operator ˆ B ψ i = b i ψ i. (16) If the operators A ˆ ad B ˆ commute, the from the Heiseberg Ucertaity Priciple the observable properties A ad B may be measured exactly, ΔA ΔB 0. (17) This result is also i accord with the Measuremet Postulates of quatum mechaics. The first Measuremet Postulate states that if a system is i a state ψ that is a eigefuctio of the operator correspodig to the property, the measuremet yields the eigevalue exactly. If we wish to measure two properties simultaeously ad exactly, the oly way we may do this is if the state ψ is a eigefuctio for both operators.

5 5 APPENDIX 1. The Mometum Represetatio We geerally express the wavefuctio of a system i terms of positio; i.e., ψ( x). However, it is possible ad sometimes desirable to covert to a represetatio i which the mometum is the variable upo which the wavefuctio depeds. The positio ad mometum represetatios of the wavefuctio are related by a Fourier Trasform, φ( p x ) = 1 π ψ( x) e ip xx /! dx. (1) The mometum space probability desity, φ( p x ), is defied as the probability desity for fidig the particle with a mometum value betwee p x ad p x + dp x. To get back to the positio represetatio, the iverse Fourier Trasform is employed, ψ( x) = 1! π φ( p x ) e +ip xx /! dp x. ()

6 6 APPENDIX. Derivatio of the Heiseberg Ucertaity Priciple There are five steps ivolved i the derivatio of the Heiseberg Ucertaity Priciple: 1. Proof of the Schwarz Iequality.. Proof of α β α β, where α ˆ ad β ˆ are geeral Hermitia operators. 3. Proof of α β * = β α, where α ˆ ad β ˆ are the same geeral Hermitia operators as i step. 4. Proof of 1 4 = b, where α β = a + i b. Also, we use step to show that α β Use of α ˆ = A ˆ A, ad β ˆ = B ˆ B to geerate the Ucertaity Priciple. 1. Proof of Schwarz Iequality We wish to prove the Schwarz Iequality, ψ φ ψ ψ φ φ, where ψ ad φ are arbitrary fuctios. To begi the proof, we defie f ( λ) = φ λψ φ λψ, (1) where λ is a parameter (that may be complex). Note that for some fuctio f(x) (which could be complex), we ca write f ( x) = a( x) + i b x ( ) i order to split f(x) ito its real ad imagiary parts. The fuctios a( x) ad b( x) are the real fuctios. The absolute square of the fuctio f(x) is defied as f ( x) = f * ( x) f ( x) = { a( x) i b( x) }{ a( x) + i b( x) } ( ) = { a( x) } + { b( x) }. f x Sice a( x) ad b( x) are real, we kow for all x that { a( x) } 0 ad { b( x) } 0.

7 7 Therefore, f ( x) 0. Because of this, we also have for ay fuctio f ( x) dx 0. Usig this relatio ad the expressio for f ( λ) from Eq. (1) above, f ( λ) = φ λψ φ λψ f ( λ) 0. ( ) * ( φ λψ) dτ = φ λψ 0 = ( φ λψ) dτ () The parameter λ is arbitrary. It ca be chose to be ay umber. To get the Schwarz Iequality, select λ = ψ φ ψ ψ. The, λ * = φ ψ ψ ψ. The expressio for f ( λ), Eq. (1), ca be expaded to yield f ( λ) = φ λψ φ λψ Substitutig the choices for λ ad λ * gives = φ φ λ * ψ φ λ φ ψ + λ * λ ψ ψ. f ( λ) = φ φ λ * ψ φ λ φ ψ + λ * λ ψ ψ = φ φ φ ψ ψ ψ ψ φ ψ φ ψ ψ φ ψ + φ ψ ψ ψ ψ φ ψ ψ ψ ψ. The last two terms cacel, which leads to f ( λ) = φ φ φ ψ ψ ψ ψ φ. (3) Sice from Eq. (), we have that f ( λ) 0, Eq. (3) ca be rewritte as φ φ φ ψ ψ ψ ψ φ 0. Rearragig,

8 8 φ φ ψ ψ φ ψ ψ φ. Sice φ ψ = ψ φ *, this equatio becomes the Schwarz Iequality, φ φ ψ ψ φ ψ ψ φ, φ φ ψ ψ ψ φ * ψ φ, φ φ ψ ψ ψ φ. Alterately, this equatio ca be writte as ψ φ φ φ ψ ψ.. Proof of α β α β, where ˆ α ad ˆ β are geeral Hermitia operators. To show this relatio, we must use the defiitio of the complex cojugate of a itegral, f O ˆ g * = ˆ O g f. Startig with the left side of the relatio that we wish to prove, we have α β = ψ α ˆ β ˆ ψ = ψ α ˆ β ˆ ψ * ψ α ˆ β ˆ ψ = α ˆ β ˆ ψ ψ ψ α ˆ β ˆ ψ. (4) Next, we use the defiitio of a Hermitia operator. If ˆ O is Hermitia, the f O ˆ g = O ˆ f g. Eq. (4) ca therefore be writte as α β = α ˆ β ˆ ψ ψ ψ α ˆ β ˆ ψ = ˆ β ψ ˆ β ψ. (5)

9 9 Oce agai, the complex cojugate of a itegral is utilized, f ˆ O g * yields = ˆ O g f. Substitutig this ito Eq. (5) α β = ˆ β ψ = α β = β ˆ ψ * α ˆ ψ ˆ β ψ. ˆ β ψ ˆ β ψ (6) Now the Schwarz Iequality ca be used, with the replacemets Substitutig, ψ α ˆ ψ, φ β ˆ φ. α β = β ˆ ψ β ˆ ψ ˆ β ψ. (7) If the operators α ˆ ad β ˆ are Hermitia, ˆ β ψ β ˆ ψ = ψ β ˆ ψ = β α ˆ ψ = ψ α ˆ ψ = α. Fially, these relatios ca be used i Eq. (7) to yield the desired result, α β ˆ β ψ ˆ β ψ or α β β α. 3. Proof of α β * = β α, where ˆ α ad ˆ β are the same geeral Hermitia operators as i step. Start with the left side, α β * = ψ α ˆ β ˆ ψ *. Use the defiitio of the complex cojugate of a itegral, f O ˆ g * = ˆ O g f, to get α β * = ψ α ˆ β ˆ ψ * = α ˆ β ˆ ψ ψ. Usig the Hermitia character of ˆ α leads to

10 10 α β * = α ˆ β ˆ ψ = ˆ β ψ ψ. Usig the Hermitia character of ˆ β leads to α β * = ˆ β ψ = ψ ˆ β. Therefore, this shows that α β * = ψ or α β * = βα. ˆ β 4. Proof of 1 4 = b, where α β = a + i b. Sice a expectatio value is just a umber (which might be complex), we defie α β = a + i b, where a ad b are real umbers. From this defiitio ad Part 3 of the Proof, βα = α β * = ( a + i b) * βα = a i b. Subtractig β α = a i b from α β = a + i b gives α β βα = a + i b ( a i b) = i b. This expressio ca be rewritte i terms of a expectatio value of the commutator, Usig the properties of itegrals, we also have The, α β βα = i b α β βα = i b [ α ˆ β ˆ ] = i b. * = i b.

11 11 [ ] = α ˆ, β ˆ * = ( i b) ( i b) = 4b. Solvig for b, b = 1 4. (8) I additio, from Part of the Proof, α β α β. The right side of this equatio ca be writte as α β = α β * α β = ( a ib) ( a + ib) α β = a + b. (9) From Part of the Proof, α β α β. From Eq. (9) we ca substitute for the right side, α β a + b. Sice a 0 ad b 0, it must be true that a + b b. Usig the expressio for b from Eq. (8), α β a + b b or α β 1 4. (10)

12 1 5. Use of α ˆ = A ˆ A, ad Let α ˆ = A ˆ A ad β ˆ = B ˆ B to geerate the Ucertaity Priciple. β ˆ = B ˆ B. The, α = ( A A ) = A A A + A = A A + A α = A A. Therefore, usig the defiitio of the quatum mechaical ucertaity, α = ( ΔA). Similarly, carryig out the same expasio for β, we ca show that β = ( ΔB). The commutator of α ˆ ad β ˆ is give by = A ˆ A, B ˆ B [ ] A, B ˆ [ ] A ˆ [, B ] A, B ˆ [ ] + A, B = ˆ [ ]. (11) Sice A ad B are just costats, they commute with ay other operators. Therefore, the last three terms i Eq. (11) are zero. The commutator becomes From Part 4 of the Proof, we have = A ˆ, B ˆ [ ]. α β 1 4. Substitutig the defiitios of α ˆ ad β ˆ, this relatio becomes ( ΔA) ( ΔB) 1 4 A ˆ [, B ˆ ]. Takig the square root of both sides gives the Heiseberg Ucertaity Priciple, ( ΔA) ( ΔB) 1 A ˆ [, B ˆ ].

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