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CRC Press eBooks

Spin and the Addition of Angular Momenta

Şaki̇r Erkoç

Published Jul 28, 2006• 0 citations
Publisher

Abstract

Systems with spin one–half are of especial interest, since this is the spin encountered in the stable particles: electrons, positrons, protons, and neutrons. (The neutron is stable only in an atomic nucleus.) In this case, the wave function is of the form 12.1 ψ ¯ = [ a 1 / 2 ( r , t ) a − 1 / 2 ( r , t ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429136528/397e01c1-e101-4a47-b552-a910642bbb8c/content/eqn12_1.tif"/>

DOI
10.1201/9781584887331-12

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Rules of Composition of Angular Momentum

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A spin 1/2 state is characterized by two amplitudes. In general a = a + (1/2) + a – (–1/2) where (1/2) stands for ( 1 0 ) , ( − 1 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq43.tif"/> for ( 0 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq44.tif"/> , and a for ( a + a − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq45.tif"/> .

CRC Press eBooks•Feb 19, 2018•0 citations

Angular Momenta

K. Kong Wan

Q36(1) The spherical harmonics Y 1,1 ( θ , φ ), Y 1,0 ( θ , φ ), Y 1, −1( θ , φ ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by 1 Y 1 , − 1 ( x , y , z ) = 3 8 π x − i y r , Y 1 , 0 ( x , y , z ) = 3 4 π z r , Y 1 , 1 ( x , y , z ) = − 3 8 π x + i y r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0667.tif"/> (a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator 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/ieq1694.tif"/> in Eq. (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 . L ^ z Y → 1 , 0 : = ( x p ^ y − y p ^ x ) Y 1 , 0 = x ( i ℏ 3 4 π z y r 3 ) − y ( i ℏ 3 4 π z x r 3 ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0673.tif"/> Finally we have 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/eqn0674.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/eqn0675.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/eqn0676.tif"/> 204 SQ36(1)(b) (1) Consider a transformation from the original coordinate system ( x , y , z ) to a new coordinate system ( x ′, y ′, z ′): 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/eqn0677.tif"/> We can see that the new z ′-axis coincides with the original x -axis, and that r = x 2 + y 2 + z 2 = x ′ 2 + y ′ 2 + z ′ 2 = r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0678.tif"/> Physically we expect the angular momentum about the x -axis to be the same as the angular momentum about the z′ -axis. We can confirm this mathematically by showing that L ^ x = y p ^ z − z p ^ y = − i ℏ ( y ∂ ∂ z − 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/eqn0679.tif"/> is equal to L ^ z ′ = x ′ p ^ y ′ − y ′ p ^ x ′ = − i ℏ ( x ′ ∂ ∂ y ′ − y ′ ∂ ∂ x ′ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0680.tif"/> This is obvious since x′ = y , y′ = z . It follows that the eigenfunctions and eigenvalues of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1700.tif"/> are the same in terms of the dashed coordinates as that of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1701.tif"/> . (2) We know the eigenfunctions Y ℓ , m ℓ ′ ( x ′ , y ′ , z ′ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1702.tif"/> and eigenvalues of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1703.tif"/> in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of L ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1704.tif"/> in the original coordinates, i.e., we have Y 1 , − 1 ′ ( x ′ , y ′ , z ′ ) = 3 8 π x ′ − i y ′ r ′ , Y 1 , 0 ′ ( x ′ , y ′ , z ′ ) = 3 4 π z ′ r ′ Y 1 , 1 ′ ( x ′ , y ′ , z ′ ) = − 3 8 π x ′ + i y ′ r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0681.tif"/> 205 Since L ^ z ′ = L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1705.tif"/> these are also eigenfunctions of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1706.tif"/> Written in terms of the original coordinates these functions become Y 1 , − 1 ′ = 3 8 π y − i z r , Y 1 , 0 ′ = 3 4 π x r , Y 1 , 1 ′ = − 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/eqn0682.tif"/> We can conclude that the required eigenfunctions L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1707.tif"/> denoted more conveniently by X 1,1 , X 1,0 , X 1, −1 are X 1 , − 1 = 3 8 π y − i z r , X 1 , 0 = 3 4 π x r , X 1 , 1 = − 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/eqn0683.tif"/> Q36(2) Suppose 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/ieq1708.tif"/> and 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/ieq1709.tif"/> are measured giving the eigenvalues 2 ħ 2 and – ħ , respectively. A measurement of 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/ieq1710.tif"/> is then made. What are the possible results of the measurement of 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/ieq1711.tif"/> ? Find the probability of each of these possible results.

•Aug 27, 2020•0 citations

Many-Particle Systems

M.S. Rogalski, Stuart B. Palmer

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CRC Press eBooks•Nov 5, 2020•0 citations

Dirac Equation

Lukong Cornelius Fai

We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.

CRC Press eBooks•Apr 19, 2022•0 citations

Electron Spin in the Dirac Theory

Lukong Cornelius Fai

In non-relativistic quantum mechanics, the angular momentum, https://www.w3.org/1998/Math/MathML"> L ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , is an integral of motion. The spin is added as a supplementary hypothesis. What should be the integral of motion in the Dirac theory for a particle described by the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ?: 18.1 https://www.w3.org/1998/Math/MathML"> Η ^ = c α ^ p ^ + β m c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>

CRC Press eBooks•Apr 19, 2022•0 citations

Time—Dependent Perturbation Theory

Şaki̇r Erkoç

Systems described by a time–independent potential energy function, V ( r , t ) = V ( r ), are the subject of quantum statics . Such systems may be studied by the time–dependent Schrödinger equation 16.1 H Ψ = i ℏ ∂ Ψ ∂ t https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429136528/397e01c1-e101-4a47-b552-a910642bbb8c/content/eqn16_1.tif"/>

CRC Press eBooks•Jul 28, 2006•0 citations
Systems with spin one–half are of especial interest, since this is the spin encountered in the stable particles: electrons, positrons, protons, and neutrons. (The neutron is stable only in an atomic nucleus.) In this case, the wave function is of the form 12.1 ψ ¯ = [ a 1 / 2 ( r , t ) a − 1 / 2 ( r , t ) ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429136528/397e01c1-e101-4a47-b552-a910642bbb8c/content/eqn12_1.tif"/>
A spin 1/2 state is characterized by two amplitudes. In general a = a + (1/2) + a – (–1/2) where (1/2) stands for ( 1 0 ) , ( − 1 / 2 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq43.tif"/> for ( 0 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq44.tif"/> , and a for ( a + a − ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429492501/e6a0c676-1f8d-4684-97a0-6bf3353f7ce0/content/eq45.tif"/> .
Q36(1) The spherical harmonics Y 1,1 ( θ , φ ), Y 1,0 ( θ , φ ), Y 1, −1( θ , φ ) are given by Eqs. (16.65) to (16.67). In Cartesian coordinates, these functions are given by 1 Y 1 , − 1 ( x , y , z ) = 3 8 π x − i y r , Y 1 , 0 ( x , y , z ) = 3 4 π z r , Y 1 , 1 ( x , y , z ) = − 3 8 π x + i y r . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0667.tif"/> (a) Using the expression in Cartesian coordinates for the quantised orbital angular momentum operator 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/ieq1694.tif"/> in Eq. (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 . L ^ z Y → 1 , 0 : = ( x p ^ y − y p ^ x ) Y 1 , 0 = x ( i ℏ 3 4 π z y r 3 ) − y ( i ℏ 3 4 π z x r 3 ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0673.tif"/> Finally we have 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/eqn0674.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/eqn0675.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/eqn0676.tif"/> 204 SQ36(1)(b) (1) Consider a transformation from the original coordinate system ( x , y , z ) to a new coordinate system ( x ′, y ′, z ′): 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/eqn0677.tif"/> We can see that the new z ′-axis coincides with the original x -axis, and that r = x 2 + y 2 + z 2 = x ′ 2 + y ′ 2 + z ′ 2 = r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0678.tif"/> Physically we expect the angular momentum about the x -axis to be the same as the angular momentum about the z′ -axis. We can confirm this mathematically by showing that L ^ x = y p ^ z − z p ^ y = − i ℏ ( y ∂ ∂ z − 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/eqn0679.tif"/> is equal to L ^ z ′ = x ′ p ^ y ′ − y ′ p ^ x ′ = − i ℏ ( x ′ ∂ ∂ y ′ − y ′ ∂ ∂ x ′ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0680.tif"/> This is obvious since x′ = y , y′ = z . It follows that the eigenfunctions and eigenvalues of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1700.tif"/> are the same in terms of the dashed coordinates as that of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1701.tif"/> . (2) We know the eigenfunctions Y ℓ , m ℓ ′ ( x ′ , y ′ , z ′ ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1702.tif"/> and eigenvalues of L ^ z ′ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1703.tif"/> in the transformed coordinates (x′, y′, z′) as they are of the same form as those of the eigenfunctions and eigenvalues of L ^ z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1704.tif"/> in the original coordinates, i.e., we have Y 1 , − 1 ′ ( x ′ , y ′ , z ′ ) = 3 8 π x ′ − i y ′ r ′ , Y 1 , 0 ′ ( x ′ , y ′ , z ′ ) = 3 4 π z ′ r ′ Y 1 , 1 ′ ( x ′ , y ′ , z ′ ) = − 3 8 π x ′ + i y ′ r ′ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/eqn0681.tif"/> 205 Since L ^ z ′ = L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1705.tif"/> these are also eigenfunctions of L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1706.tif"/> Written in terms of the original coordinates these functions become Y 1 , − 1 ′ = 3 8 π y − i z r , Y 1 , 0 ′ = 3 4 π x r , Y 1 , 1 ′ = − 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/eqn0682.tif"/> We can conclude that the required eigenfunctions L ^ x https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429296475/bdc73243-97dc-4cf5-9624-109e4c3275b1/content/ieq1707.tif"/> denoted more conveniently by X 1,1 , X 1,0 , X 1, −1 are X 1 , − 1 = 3 8 π y − i z r , X 1 , 0 = 3 4 π x r , X 1 , 1 = − 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/eqn0683.tif"/> Q36(2) Suppose 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/ieq1708.tif"/> and 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/ieq1709.tif"/> are measured giving the eigenvalues 2 ħ 2 and – ħ , respectively. A measurement of 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/ieq1710.tif"/> is then made. What are the possible results of the measurement of 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/ieq1711.tif"/> ? Find the probability of each of these possible results.
The postulates of quantum mechanics and their consequences, which have been developed to describe a single particle, are still valid when applied to a system of N particles. The system can be described by a wave function Ψ which depends on the time and on the 3 N coordinates of all the particles: 10.1 Ψ = Ψ ( r → 1 , … r → i , … r → N , t ) = Ψ ( x 1 , y 1 , z 1 , … , x i , y i , z i , … x N , y N , z N , t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn10_1_B.tif"/>
We re-examine the square of the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , in Equation 15.7 and also the Dirac approach in finding the relativistic Hamiltonian. The classical Hamiltonian for the free particle should be the positive square root of Equation 15.7 which is not a linear function of the momenta. Applying this to the problem, say for example in Equation 15.10, we have the resultant wave equation that is asymmetrical with respect to space and time derivatives and so the necessity to linearize the equation in the momentum operators. So for Equation 15.10, it is impossible to include the Pauli spin matrices without destroying the invariance of the theory since the spin matrices transform the components of a three-dimensional rather than a four-dimensional vector. In addition, the function, https://www.w3.org/1998/Math/MathML"> Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , has one component rather than two components like spinors and so the Schrödinger relativistic equation represents a particle with no spin. So for a potential to be added in Equation 15.10, the Lorentz-transformation properties of such an addition should be guaranteed by matrix algebra which is the requirement for the Hamiltonian https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math16_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> to be linear in the momentum operators and guarantees relativistic invariance. This matrix algebra which permits the linearization of the Hamiltonian in the momentum operator was introduced by Dirac with the help of the Pauli matrices.
In non-relativistic quantum mechanics, the angular momentum, https://www.w3.org/1998/Math/MathML"> L ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_1.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> , is an integral of motion. The spin is added as a supplementary hypothesis. What should be the integral of motion in the Dirac theory for a particle described by the Hamiltonian, https://www.w3.org/1998/Math/MathML"> Η ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_2.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/> ?: 18.1 https://www.w3.org/1998/Math/MathML"> Η ^ = c α ^ p ^ + β m c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003273073/d236e21f-f9f6-4e42-b186-ef1c50f0a68b/content/math18_3.tif" xmlns:xlink=" https://www.w3.org/1999/xlink "/>