CRC Press eBooks

Matrix Aspects of Quantum Mechanics

F.J. Duarte

Published Oct 29, 2013• 0 citations

Abstract

As mentioned in Chapter 1 , the Heisenberg matrix mechanics provides one of the three main avenues to quantum mechanics. This approach to quantum mechanics was disclosed in three papers authored by Heisenberg (1925) , Born and Jordan (1925) , and Born et al. (1926) . An iconic result from the Heisenberg–Born–Jordan contribution was the commutation rule ( 14.1 ) p q − q p = h 2 π i https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315216720/70121b16-f2c6-4331-b1ba-eecb806b93da/content/eq595.tif"/>

DOI: 10.1201/b16055-14

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Angular Momenta

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(27.86), verify by explicit calculations that Y 1,1 ( x , y , z ), Y 1,0 ( x , y , z )and Y 1,-1 ( x , y , z ) are the eigenfunctions of L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1695.tif"/> corresponding to eigenvalues ħ , 0 and – ħ . (b) Consider the following coordinate transformations: x → z ′ , y → x ′ , z → y ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0668.tif"/> (1) 202 Show that the component of the orbital angular momentum operator along the z ′-direction is the same as that along the x -direction, i.e., show that L ^ c z ′ = L ^ c z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1696.tif"/> . (2) Show that the simultaneous eigenfunctions of L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1697.tif"/> and L ^ c x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1698.tif"/> corresponding to eigenvalues of L ^ c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1699.tif"/> equal to 2 ħ 2 are given in Cartesian coordinates by X 1 , − 1 ( x , y , z ) = 3 8 π y − i z r , X 1 , 0 ( x , y , z ) = 3 4 π x r , X 1 , 1 ( x , y , z ) = − 3 8 π y + i z r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0669.tif"/> SQ36(1)(a) In Cartesian coordinates we have 2 : p ^ x Y → 1 , − 1 : = − i ℏ 3 8 π ∂ ∂ x x − i y r = − i ℏ 3 8 π ( 1 r − x − i y r 2 ∂ ∂ x r ) = − i ℏ 3 8 π ( 1 r − x − i y r 2 x r ) = − i ℏ 3 8 π 1 r 3 ( r 2 − x 2 + i x y ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0670.tif"/> p ^ y Y → 1 , − 1 : = − i ℏ 3 8 π ∂ ∂ y x − i y r = − i ℏ 3 8 π ( − i r − x − i y r 2 ∂ ∂ y r ) = − i ℏ 3 8 π ( − i r − x − i y r 2 y r ) = − i ℏ 3 8 π 1 r 3 ( − i r 2 − x y + i y 2 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0671.tif"/> L ^ z Y → 1 , − 1 : = ( x p ^ y − y p ^ x ) Y 1 , − 1 = − i ℏ 3 8 π 1 r 3 ( ( − i x r 2 − x 2 y + i x y 2 ) − ( y r 2 − y x 2 + i x y 2 ) ) = − i ℏ 3 8 π 1 r 3 ( − i x r 2 − y r 2 ) = − ℏ 3 8 π 1 r { x − i y } = − ℏ Y 1 , − 1 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0672.tif"/> 203 Next we have p ^ x Y → 1 , 0 : = − i ℏ 3 4 π ∂ ∂ x z r = − i ℏ 3 4 π ( − z r 2 ∂ ∂ x r ) = i ℏ 3 4 π z x r 3 . p ^ y Y → 1 , 0 : = − i ℏ 3 4 π ∂ ∂ y z r = − i ℏ 3 4 π ( − z r 2 ∂ ∂ y r ) = i ℏ 3 4 π z y r 3 . 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Quantum gravity with photon interactions, cavity resonators with inhomogeneities, classical and quantum optimal control of fields

Harish Parthasarathy

We wish j t ( X ) to track the noiseless trajectory X d ( t )∈ L ( h ). Assume non-demoltion measurements Y o ( t )= U ( t )* Y i ( t ) U ( t ) are made and the Belavkin filter for π t ( X ) =  j t ( X ) | η t ) , η t = σ ( Y 0 ( s ) : s ≤ t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1006.tif"/> has been constructed as d π t ( X ) = F t ( X ) d t + G t ( X ) d Y o ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1007.tif"/> trajectory estimation error X d ( t )− π t ( X ) is given as feedback to the state equations in the form d j t ( X ) = j t ( θ 0 ( X ) ) d t ⁢ + ⁢ j t ( θ 1 ( X ) ) d A ( t ) + j t ( θ 2 ( X ) ) d A ( t ) * + ⁢ j t ( θ 3 ( X ) ) d Λ ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1008.tif"/> 192 + ⁢ K ( t ) ( X d ( t ) − π t ( X ) ) d t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1009.tif"/> Alternately, if we assume that X d follows a noiseless trajectory, ie, it evolves according to the equation X d ( t ) , = j t ( 0 ) ( X d ) = U 0 ( t ) * X d U 0 ( t ) , U 0 ( t ) = e x p ( − i t H 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1010.tif"/> then we can give as feedback the estimation error E ( t ) = X d ( t ) − T r 2 ( π t ( X ) ( I ⊗ | ϕ ( u ) > < ϕ ( u ) | ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1011.tif"/> which is a system observable. This error feedback is given to the above state equations, ie, to the Evans-Hudson flow resulting in the dynamics d j t ( X ) = j t ( θ 0 ( X ) ) d t + j t ( θ 1 ( X ) ) d A ( t ) + j t ( θ 2 ( X ) ) d A ( t ) * + j t ( θ 3 ( X ) ) d Λ ( t ) + K ( t ) E ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1012.tif"/> Alternately, we can incorporate this error feedback in the original HP equation as d U ( t ) ⁢ = ⁢ ( - ( i H + ⁢ P + ⁢ K ( t ) E ( t ) ) d t + L 1 d A ( t ) ⁢ + ⁢ L 2 d A ( t ) * ⁢ + ⁢ S d Λ ( t ) ) U ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1013.tif"/> resulting in the Evans-Hudson flow the same as above but with θ 0 (X) replaced by θ 0 ( X )+ iK ( t )[ E ( t ), X ]. Luc-Bouten’s method of control is slightly different. Here, we choose a system observable Z and give an infinitesimal control unitary U c ( t , t ⁢ + ⁢ d t ) ⁢ = ⁢ U ( t ⁢ + ⁢ d t ) * e x p ( - i Z d Y i ( t ) ) U ( t + d t ) ⁢ = ⁢ e x p ( i Z ( t + d t ) d Y o ( t ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1014.tif"/> Note that Y o ( t ) ⁢ = ⁢ U ( t + d t ) * Y i ( t ) U ( t + d t ) , Y o ( t + d t ) ⁢ = ⁢ U ( t + d t ) * Y i ( t + d t ) U ( t + d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1015.tif"/> and hence, forming the difference, we get d Y 0 ( t ) = U ( t + d t ) * d Y i ( t ) U ( t + d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1016.tif"/> Also, Z commutes with dY i ( t ) and hence Z ( t + dt )= U ( t + dt )* ZU ( t + dt ) commutes with U ( t + dt )* dY o ( t ) U ( t + dt ). Let ρ c ( t ) denote the controlled state at time t, ie after applying the Belavkin filter and control upto time t to the HP evolved state. Then, we apply the Belavkin filter from t to t + dt , via ρ B ( t + d t ) - ρ c ( t ) + δ ρ B ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1017.tif"/> where δ ρ B ( t ) = L t * ( ρ c ( t ) ) d t + ( M t ρ c ( t ) + ρ c ( t ) M t * - T r ( M t + M t * ) ρ c ( t ) ) ρ c ( t ) ) ( d Y o ( t ) - T r ( ρ c ( t ) ( M t + M t * ) ) d t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1018.tif"/> Finally, the filtered and controlled state at time t + dt is given by ρ c ( t + d t ) - ρ c ( t ) ⁢ + ⁢ δ ρ c ( t ) = https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003163626/7a1b5c8e-5b02-4966-b7fd-03fb766469c4/content/math1019.tif"/>

CRC Press eBooksFeb 3, 20210 citations

A glimpse of matrix mechanics via the harmonic oscillator

Elina Palmgren, Ricardo Karam

In this paper, we provide a glimpse of the original formulation of quantum mechanics (a.k.a. matrix mechanics), as proposed by Heisenberg, Born, and Jordan in 1925, by showing how it solves the quantum harmonic oscillator. This presentation can shed light on the relationship between different formulations and give students a deeper understanding of quantum mechanics.

American Journal of PhysicsDec 23, 20240 citations