Foundations of Complex Mechanics

Transcription

1 Foundations of Compex Mechanics Chapter Outine Quantum Hamiton Mechanics Dynamica Representation of Quantum State Compex Variabes and Quantum Operators Orbita and Spin Anguar Momentum Compex Hamitonian and Compex Energy Chapter Summary Probems Quantum mechanics had been estabished from the foowing two postuates: () Postuate of correspondence: to any sef-consistent and we-defined observabes A, there corresponds an operator Â. () Postuate of quantization: The operator  corresponds to the observabe A( q, p ) can be constructed by repacing the coordinate q and momentum p in the expression for A by the assigned operators q qˆ = q and p ˆp = iħ. The vaidity of the two postuates was justified indirecty via the vouminous precise predictions of quantum mechanics. Athough the two postuates work very successfuy, unti now, we sti do not know why they shoud work and a forma proof of their origin sti acks. We aso do not know very ceary about the underying reason that to obtain the correct operators in coordinate system other than Cartesian coordinates, it is aways necessary to transform A( q, p ) into Cartesian coordinates before putting in the operators. This chapter aims to prove the two postuates by the first principe of Hamiton mechanics, to expound the reason why the postuate of quantization is ony true in the Cartesian coordinates, to demonstrate how to obtain directy quantum operators in spherica coordinates without transforming back to Cartesian coordinates.. 3. Quantum Hamiton Mechanics The main idea of the compex mechanics is based on the equivaence between a compex observabe A( q, p ) in Hamiton mechanics and its associated operator  in quantum mechanics, where ( q, p ) are canonica variabes defined in compex domain. Based on this equivaence and the requirement that the behavior of A( q, p ) must obey Hamiton equations, we can determine the expression and the various quantization properties of  directy from Hamiton equations of motion, regardess of the coordinate system being used. The Hamiton equations considered here are derived from a quantum Hamitonian H, which is different from the cassica one. To find out the correct quantum Hamitonian, we first reca a cassica resut that for a given cassica Hamitonian Hc( t, q, p ), the cassica Hamiton-Jacobi (H-J) equation reads Sc + Hc( t, q, p ) =0, (3..) p= Sc t where S c is the cassica action function. We may regard the cassica H-J equation as the short waveength imit of Schrodinger equation (Godstein, 980): ħ iħ = + V, (3..) t m 56

2 as can be seen via the foowing transformation from which Schrodinger equation becomes = exp(i S / ħ ), (3..3) S iħ + ( S) + V = S. (3..4) t m m We recognize the quantity in brackets as the cassica Hamitonian for a singe partice described in Cartesian coordinates. Eq.(3..4) is known as the quantum H-J equation, which reduces to the cassica H-J equation (3..) if the right-hand side of Eq.(3..4) is negigibe, which means that the waveength of the matter wave is so short that the momentum changes by a negigibe fraction over a distance of waveength (Godstein, 980). The transformation (3..3) was first introduced by Schrodinger in transforming the phase function φ= S / h governed by Fresne s wave equation to the wavefunction governed by Schrodinger s wave equation. If we treat Eq.(3..4) as the quantum-mechanica counterpart of the cassica H-J equation (3..), it is natura to ask what wi be the corresponding quantum-mechanica counterpart of the cassica Hamitonian H. Rewriting Eq.(3..4) in a form anaogous to Eq.(3..): c S + Ht (, q, p ) =0, (3..5) p= S t we obtain the desired quantum Hamitonian H, compatibe with Schrodinger equation, as H( ) = p + V( q) + Q( ( q )), (3..6) m where Q is known as quantum potentia defined by ħ ħ ħ Q( ( q)) = p= S= n ( q ). (3..7) mi mi m The usage of the notation H ( ) is to emphasize the state-dependent nature of the quantum Hamitonian H. The cassica Hamitonian H c depends ony on the externay appied potentia V( q ), whereas the quantum Hamitonian H ( ) depends on V( q ) as we as on the interna state where the partice ies. For a given state described by ( q ), the quantum Hamitonian H ( ) defined in Eq.(3..6) is an expicit function of the canonica variabes q and p that are regarded as independent variabes. There are two roes payed by the wavefunction in the quantum Hamitonian H. Firsty, as indicated in Eq.(3..5), it determines the canonica momentum p j according to S n pj= = iħ. (3..8) qj qj Secondy, it generates the quantum potentia Q according to Eq.(3..7). The equations of motion for a partice moving in the quantum state are derived by appying the quantum Hamitonian H to the Hamiton equations dq H( ) = = p, (3..9a) dt p m d p H( ) ħ = = V( q) n ( q). (3..9b) dt q q m Note that as in cassica Hamiton mechanics, we have obtained H / p and H / q by treating q and p as independent variabes in the Hamitonian H in Eq.(3..6). The Hamiton equations (3..9) are distinct from the cassica ones in two aspects: the compex nature and the state-dependent nature. The compex nature is a consequence of the fact that the canonica variabes ( q, p ) soved from Eq.(3..8) and Eqs.(3..9) are, in genera, compex variabes. The state-dependent nature means that the Hamiton equations of motion (3..9) govern the quantum motion excusivey in the specific quantum state described by. We wi show in the subsequent sections and chapters that a the quantum operators and the various quantum effects can be derived from the compex canonica 57

3 momentum in Eq.(3..8) and from the compex-extended Hamiton equations of motion in Eqs.(3..9). In the next section, we first discuss the meanings of a wavefunction in Hamiton mechanics by pointing out that each wavefunction represents a dynamic system described by Eqs.(3..9). Hence, information contained in can be extracted from the dynamic system (3..9) using methods deveoped in anaytica mechanics. Section.3 estabishes the reationship between a compex observabe A( q, p ) in Hamiton mechanics and its associated operator  in quantum mechanics. This reationship aows us to derive any operator  from its counterpart compex function A( q, p ) in Hamiton mechanics and to express the commutator [ AB, ] in terms of the Poisson bracket { AB, } in Hamiton mechanics. Hamiton equations of motion (3..9) are vaid ony for Cartesian coordinates. In Section.4, we derive quantum operators and Hamiton equations of motion in spherica coordinates, based on which we wi aow us to sove quantum centra-force probems, such as the eectron motion in hydrogen atom, in Chapter 9. Quantum operators expressed in genera curviinear coordinates and in the presence of eectromagnetic fied are derived from the Hamiton equations of motion in chapter. 3. Dynamica Representation of Quantum State In the framework of quantum Hamiton mechanics, a wavefunction is not merey an abstract function in an infinite-dimensiona vector space; it aso represents a concrete dynamic system. Definition 3.. The quantum state assigned by a wavefunction is a dynamic system whose phase-space trajectory ( q( t), p ( t)) obeys the Hamiton equations (3..9) with the Hamitonian H given by Eq.(3..6). According to this definition, when we say that a quantum observabe A( q, p ) is evauated in the state 0, it actuay means that A( q, p ) is evauated aong a phase-space trajectory ( q( t), p ( t)) determined from Eqs.(3..9) with H specified by 0. On soving ( q( t), p ( t)) from Eqs.(3..9), it can be shown as in the foowing theorem that the soution for p is aready given by Eq.(3..8); that is to say, the soution of Schrodinger equation,, provides the first integration of the Hamiton equations (3..9). Theorem 3... For a given soution of Schrodinger equation, the quantum momentum p determined from Eq.(3..8) and the quantum potentia Q ( ( q )) determined from Eq.(3..7), satisfy automaticay the quantum Newton equation (3..9b), i.e., H ṗ= = ( V+ Q). (3..) q q Proof: We sha start with Eq.(3..5) and show that it eads directy to Eq.(3..9b). The tota differentiation of Eq.(3..5) with respect to the Cartesian coordinates q = ( q, q,, q N ) reads d S + H =0, i=,,, N dqi t, (3..) where with S= St (, q ), we have d( S / t)/ dqi= S / qi t, whie with H= Ht (, q, p( q )), we have N dh H H pk = + dq q, (3..3) p q i i k= k i with p k given by Eq.(3..8) as pk= S / qk. Assuming that S is twice continuousy differentiabe with respect to q k, we may rewrite pk / qi as pk S S pi = = =. (3..4) q q q q q q i i k k i k Inserting the above identity and Eq.(3..9a) into Eq.(3..3) yieds 58

4 from which Eq.(3..) becomes N pi = + q k i i q k= k dh H dq q N i q k. k= k i pi p H + = t q q This is just Eq.(3..9b) by noting that the eft-hand side is equa to dp i / dt, the tota differentiation of p i with respect to time t. The inverse of Theorem 3.. is aso true, i.e. starting with Hamiton equations (3..9) and assuming the soution of p in the form of Eq.(3..8), we can show that the to-be-determined functions S and satisfy the quantum H-J equation (3..5) and the Schrodinger equation (3..), respectivey. Exampe 3.. As an iustrating exampe of Theorem 3.., we consider the dynamic systems corresponding to the quantum states of harmonic osciator. The Hamitonian (3..6) with V= Kx / has the form ħ d H( n ) = p + Kx n n( x). (3..5) m mdx The eigenfunction nx ( ) for harmonic osciator is found to be αx / ( x) = CH ( αxe ), n=0,,,, (3..4) n n n where α= mk / ħ and H n is the n th-order Hermite poynomia. The dynamic system associated with nx ( ) is represented by the foowing Hamiton equations: dx H( n ) p = =, (3..7a) dt p m dp H( n ) d d ħ d = = Kx n n ( x) = ( V+ Q) dt x dx mdx. (3..7b) dx Substituting Eq.(3..7a) into Eq.(3..7b) yieds, dx dv dq m =. (3..8) dt dx dx This equation has the form of Newton s second aw, in which the partice is subjected to a quantum force dq / dx in addition to the cassica force dv / dx. Treating p as a function of x, we can recast dp / dt into the form dp dpdx pdp d p = = =. (3..9) dt dxdt mdx mdx Combination of Eq.(3..7b) and Eq.(3..9) yieds Hxp (, ) = p + Vx ( ) + Q= E= constant, (3..0) m where Q is the quantum potentia present in the state n ħ d Q= mdx n ( x). (3..) Accordingy, we can regard Eq.(3..0) as the energy conservation aw for one-dimensiona quantum Hamiton system. By substituting the reation n d i dn( x) p= iħ n n( x) = ħ (3..) dx dx from Eq.(3...8) into Eq.(3..0), it turns out that Eq.(3..0) is just the time-independent Schrodinger equation for harmonic osciator n 59

5 ħ dn + ( E Vx ( )) n=0. (3..3) m dx The equivaence among the Hamiton equations (3..7), the energy conservation aw (3..0) and the Schrodinger equation (3..3) indicates that the wavefunction n actuay represents a dynamic system whose behavior obeys the Hamiton equations (3..7), as stated in Definition.. The constant C n in Eq.(3..4) has an important roe in normaizing the wavefunction n, but as can be seen from Eq.(3.,), the dynamic representation of n is independent of C n. An aternative expression of Eq.(3..0) may be obtained by repacing n with p via the reation (3..), ħdp Hxp (, ) = p + Vx ( ) + = E, (3..4) m m idx which was known as Riccati equation in the mathematica iterature. Quantum Hamiton-Jacobi theory (Leacock and Padgett, 983; Bhaa, Kapoor, and Panigrahi, 997), which was deveoped from Eq.(3..4), permits the exact determination of the bound-state energy eves and the reated eigenfunctions without the necessity of soving the corresponding Schrodinger equation. The quantum potentia Q has a cose reation to the probabiity interpretation of standard quantum mechanics; it expains the underying reason why some ocations are hard to access, whie some are accessibe with arge probabiity. Theorem 3.. The tota potentia VTota= Q+ V is inversey proportiona to the probabiity density function in the manner that the ocations with zero probabiity are where the tota potentia approaches infinity and the ocations with maximum probabiity are where the tota potentia arrives at its minimum. Moreover, the ocations with maximum probabiity are just the equiibrium points of the dynamic system (3..9) representing the quantum state. Proof: From the energy conservation aw of Eq.(3..0), we can express the tota potentia as ħ d n VTota= Q+ V= E p = E+ m m dx. (3..5) A succinct expression of quantum Newton equation (3..8) then turns out to be dvtota ħ d d n. (3..6) mx= = dx mdx dx Since this equation of motion is independent of the constant E, we can choose E=0 as the reference energy eve for V Tota. The magnitude of the tota potentia barrier now becomes ħ d n d/ dx VTota ħ = = m dx, (3..7) m which states that the height of the tota potentia barrier is inversey proportiona to. A spatia point with arge vaue of corresponds to the ocation of ow potentia barrier and hence arge accessibiity to this point. This fact egitimates the use of as the probabiity measure for a partice to appear at a specified spatia point. Besides the probabiity information provided by, the detaied trajectory under the action of V Tota can be found by integrating Eq.(3..6) whose first integration, as has been shown in Theorem 3.., is given by Eq.(3..8) and Eq.(3..9a): Fig.3.. is an iustration of Theorem 3.. by taking harmonic osciator as an exampe. The tota potentia V Tota and the probabiity density are potted together for quantum states n=0,,, and 5. The inverse proportionaity between V Tota and is ceary dispayed, from which the positions with zero probabiity are justified by the presence of infinity potentia and the 60 dx p i d/ dx = = ħ. (3..8) dt m m The equiibrium point of the above noninear system is the position having the property of x=0, i.e., d / dx=0, which in turn is the necessary condition that the probabiity density achieves its maximum.

6 positions with the maximum probabiity are justified by the owermost points of the potentia. (a) n=0 (b) n= V Tota V Tota 0 0 (c) n= (d) n=5 V Tota V Tota 5 5 Fig.3.. The iustration of the inverse proportionaity between the probabiity density and the tota potentia VTota = V+ Q ( ) for harmonic osciator with quantum states n=0,,, and 5. The ocations with zero probabiity are where the tota potentia approaches infinity and the ocations with maximum probabiity are where the tota potentia arrives at its minimum. 3.3 Compex Variabe and Quantum Operator Besides the state-dependent property, quantum Hamitonian mechanics is distinct from cassica Hamitonian mechanics in the unique feature that the observabes appeared in quantum Hamiton mechanics, such as q, p, and H, are in genera compex-vaued. For instance, if we determine the momentum p from Eq.(3..8) for a given wavefunction, we shoud find that p has rea component as we as imaginary component. This compex-vaued nature inherits from Schrodinger equation, which produces the compex-vaued wavefunction that, in turn, eads to the compex-vaued quantum Hamitonian H and quantum potentia Q. It is this compex-vaued nature that aows us to derive the correct quantum operator accompanying each quantum observabe. In the foowing, we first introduce the definition of quantum operator in Hamiton mechanics, and then demonstrate how the commony used quantum operators in Cartesian coordinates can be derived from this definition. Operators in curviinear coordinates wi be considered in the ater sections. Definition 3.3. For a quantum observabe A evauated in the quantum state, its associated quantum operator  is defined via the reation A= A ˆ. (3.3.) Using this definition, we can give a forma proof of the quantization axiom p iħ that governs the critica transition from cassica systems to quantum-mechanica systems. Theorem 3.3. (Yang, 007A) 6

7 The canonica operators ( q, p ) corresponding to the canonica variabes ( q, p ) are given by qˆ= q and ˆp = iħ, whereas the Hamitonian operator Ĥ corresponding to the quantum Hamitonian defined in Eq.(3..6) is given by H= p / m+ V. Proof: Rewriting Eq.(3..8) in the form of Eq.(3.3.), we obtain p = S= iħ n = ( / ) ( i ħ ). (3.3.) The comparison of the above equation with the definition p= ( / ) ˆp gives ˆp = iħ. As for ˆq, we may express q as q= ( / ) q and contrast this with the definition q= ( / ) q ˆ to obtain qˆ= q. To derive the Hamitonian operator Ĥ from the definition H= ( / ) Hˆ, we need to express the quantum Hamitonian H in terms of the wavefunction. The insertion of p = iħ n in Eq.(3..6) yieds ħ H ( i n) n V ħ = ħ + = + V m m m. (3.3.3) In comparison with Eq.(3.3.), Eq.(3.3.3) produces the Hamitonian operator H = p / m+ V. The canonica momentum operator ˆp derived in Theorem 3.3. must not be confused with the mechanica momentum operator ˆP. In Cartesian coordinates, the mechanica momentum P is given by P= mq and from Eq.(3..9a) we find p= mq = P, which indicates Pˆ = ˆp = iħ in Cartesian coordinates. However, in curviinear coordinates, p and P are, in genera, different and the quantization axiom P ˆ= iħ is no onger vaid as wi be expounded further in ater chapters. Eq.(3.3.) indicates that defining quantum momentum p = S in compex domain is necessary to resut in the correct momentum operator ˆp. A simiar but different quantum momentum was proposed by Bohm (95) in the form of p B= SB, where S B is the phase of the wavefunction i S / defined by B ħ =Re B with R B and S B being rea functions. If we foow the same procedures eading to Eq.(3.3.) but empoy the rea quantum momentum p B= SB instead of the compex momentum p = S, we sha find that it is not possibe to arrive at the correct momentum operator ˆp = iħ. A natura outcome of defining canonica variabes ( q, p ) in compex domain is the quantization of action variabe, a postuate proposed by Sommerfed (95) and Wison (95), Ji= pdq i i= nh, n=0,,,, i=,,, N, (3.3.4) ci where c i is a cosed trajectory in the compex q i pane obtained from the integration of the Hamiton equations (3..9) and the compex momentum p i is given by Eq.(3..8) as pi= iħ n / qi. To prove the quantization rue (3.3.4), we define the foowing conforma mappings from the q i compex pane to the compex pane: = ( q ) : q, i=,,, N (3.3.5) i i i where ( q i ) is a function of the singe compex variabe q i obtained by fixing other coordinates in the wavefunction ( q,, qi,, qn ). The function i maps a cosed path c i in the q i pane into a cosed path c i in the pane. Counting the number of encircement of the origin in the pane by the cosed path c i provides us with the quantum number n in Eq.(3.3.4). Theorem 3.3.: (Yang, 006D) Let c i be any cosed compex trajectory traced out by the coordinate q i. Then the contour integra defined in Eq.(3.3.4) over the contour c i is quantized and the reated quantum number n is equa to the number of encircement of the origin in the pane by the cosed path c i obtained from c i via the mapping i. Proof: With the substitution pi= iħ n / qi, the action variabe J i becomes ħ n ħ d ni ħ Ji= pdq i i= dqi dqi d ni c i = = c q i c dq i, (3.3.6) c i i i i i i where in the ast equaity we have expressed J i in terms of the net change of n i aong the cosed i path c i. By expressing i in a poar form i = ie θ, we can further simpify J i as 6

8 ħ J = d n + ħ dθ i. (3.3.7) i i ci ci The first term in the right-hand side is zero because the net change of n i is zero aong any cosed path. The second term is reevant to the net phase change of i aong the cosed path c i, which must be an integra mutipe of π. Therefore, Eq.(3.3.7) is reduced to the expected resut Ji= ħ dθ= ħ( nπ) = nh, ci where n is the number of encircement of the origin in the i pane by the cosed path c i and nπ represents the corresponding net phase change. Fig.3.3. iustrates the mapping between the contour c i in the compex pane q i and the contour c i in the compex i pane. Observing the number of encircement of the origin by the cosed path c i aows us to identify the quantum number n graphicay. (a) Trajectory in q pane C D Im( q ) B A Re( q ) (b) Zero encircement of the origin Im( ) C D E B A F Re( ) E F Contour c traced out by a partice (c) One encircement of the origin Contour c is the mapping of c into the pane. (d) Two encircements of the origin B Im( ) Im( ) A Re( ) C E B Re( ) C F F D D E A Fig.3.3. Conforma mapping of the cosed contour c in the q pane into the cosed contour c in the pane. The contour c is the partice s trajectory soved from quantum Hamiton equations of motion and the imagine contour c is obtained via the transformation = q ( ). As a representative point qt ( ) traces out the entire contour c in the countercockwise direction A B C D E F, its image point in the pane traces out the contour c in the direction A B C D E F. The number n of the countercockwise encircements of the origin of the pane is equa to 0,, and, respectivey, for the case (b), (c), and (d). The quantization rue (3.3.4) is independenty vaid for any coordinate qi( t ) that has periodic motion and thus has cosed trajectory in the compex q i pane, regardess of whether the whoe quantum system is periodic or not. We sha revisit Theorem 3.3. in Chapter 4 for the quantization of harmonic osciator and in Chapter 9 for the quantization of hydrogen atom. In Theorem 3.3. and Theorem 3.3., we have witnessed the necessity of extending quantum observabes such as q and p to compex domain. It is worth noting that the reation between the compex quantum observabe A and the quantum operator  in Eq.(3.3.) is in the form of strict equaity, but not merey an 63

9 abstract correspondence. In the standard approach, we obtained Ĥ by appying the abstract corresponding principe of repacing the momentum p in the cassica Hamitonian Hc= p / m+ V with the momentum operator ˆp, but in doing so we coud not estabish any equaity between Ĥ and H c. However, under the framework of quantum Hamitonian mechanics, we have a quantum Hamitonian H in Eq.(3..6), which is directy reated to Ĥ via the equaity H= ( / ) Hˆ. For any given observabe A defined in quantum Hamiton mechanics, we can identify its accompanying operator  by using Eq.(3.3.); conversey, for a given operator ˆB, an expicit expression for its accompanying quantum observabe B is found to be B = ( / )Bˆ. To famiiarize us with this equivaence, et us consider a textbook exampe regarding the operator of the anguar momentum L= q p. Exampe 3.3. In quantum Hamiton mechanics, the expression for L is the same but with q = [ xyz] and p = [ px py pz ] satisfying the quantum Hamiton equations (3..9), instead of the cassica Hamiton equations. Evauating the x components of L with p given by Eq.(3..8), we obtain n n i Lx ypz zpy y ħ z ħ ħ = = y z = i z i y z y. (3.3.8) Comparing the above equation to the definition L ( / ) ˆ x= Lx gives L ˆx as: ˆ Lx = iħ y z = ypˆ z zp y z y, (3.3.9) where the expressions for p ˆy and p ˆz has been derived in Theorem We recognize that an expicit expression of L x in terms of the wavefunction naturay eads to the expression for L ˆx. The other two components L ˆ y= zpˆ x xp z and L ˆ z= xpˆ y yp x can be derived in a simiar way. There exists a specia wavefunction n such that Eq.(3.3.) yieds A( p, q ) = A n = ( / )Aˆ n n = constant. In such a case, the observabe A( p, q ) becomes a constant in the state n. In conjunction with n, a remarkabe ink can be estabished between the conservation aw in quantum Hamiton mechanics and the concept of stationary observabe in quantum mechanics. Lemma 3.3. A quantum observabe A( p, q ) is stationary in the quantum state n, if and ony if A( p, q ) is conservative aong any phase-space trajectory ( q( t), p ( t)) determined from Eqs.(3..9) with the Hamitonian H induced by n ; furthermore, this conserved vaue of A( p, q ) is just equa to the eigenvaue of  with respect to the eigenfunction n. Proof: In quantum mechanics, an observabe A is said to be stationary in the state n, if its reated operator  satisfies  n = A nn, where A n is the eigenvaue of  corresponding to the eigenfunction n. Now we can appy Eq.(3.3.) to evauate A in the state n as A( q, p ) = ( / )Aˆ n n = ( / n )Ann = An, which states that the vaue of A( p, q ) evauated in the state n is a constant equa to the eigenvaue A n. According to Definition 3., the constancy of A( q, p ) in the state n amounts to the conservation of A( p, q ) aong any phase-space trajectory ( q( t), p ( t)) determined from Eqs.(3..9) with the Hamitonian H induced by n. Conversey, if we are given that A is a constant A n in the state n, then Eq.(3.3.) impies A= A ( / )Aˆ n= n n, i.e.,  n = A nn, which ensures that A is stationary in n. As a demonstration of Lemma.3., we consider A = H and assume that H is stationary in n, i.e., Hˆ n = E n n. If we appy this stationary condition to Eq.(3.3.), we obtain H( q, p ) = ( / ) ˆ nhn = ( / n ) Enn= En, showing that the Hamitonian H( q( t), p ( t)) in Eq.(3..6) is conservative aong any phase-space trajectory ( q( t), p ( t)) in the state n. We may confirm the conservation of H by showing dh / dt=0 in the state n. Because q ( t) and p ( t) in the state n satisfy the Hamiton equations (3..9), we obtain the expected resut dh Hdq Hd p H H H H = + = =0. (3.3.0) dt q dt p dt q p p q 64

10 Exampe 3.3. (Yang, 006A) Continue the discussion of harmonic osciator in Exampe 3.. and consider the energy conservation in the ground state, for which the wavefunction is given by CHe αx 0= 0 0. Inserting 0 into Eq.(3..) and Eq.(3..) yieds, respectivey, the quantum potentia Q= ħω / = ( ħ / ) K / m and the quantum momentum p= iħ αx. The tota energy in the ground state is then found to be a constant equa to p ħω ħω H= + Kx + Q= ħ αx + Kx + = = E0, (3.3.) m m where E0= ħ ω / is just the eigenvaue corresponding to 0. By a simiar way, we can show in the th n eigenstate n, the tota energy H is a constant equa to the eigenvaue corresponding to n, i.e., H= ( n+/ ) ħ ω= En. On the other hand, when is not an eigenfunction of Â, we have A( q, p ) = ( / )Aˆ constant with its vaue being varying with ( q( t), p ( t)). In such a case, quantum mechanics says that A is uncertain in the state and suggests adopting as the probabiity density function to extract the statistica properties of A. Quantum Hamiton mechanics provides us with an aternative way to evauate a nonstationary observabe; we may empoy the expression A( q, p ) = ( / )Aˆ to expicity trace the variation of A( q( t), p ( t)) aong any phase-space trajectory ( q( t), p ( t)) in the dynamic system (3..9) prescribed by. The equivaence estabished in Lemma 3.3. can be eucidated more concisey in terms of the equaity (not merey a correspondence) between the commutator defined in quantum mechanics and the Poisson bracket defined in Hamiton mechanics. Given two operators A and B, their commutator is defined as [ AB, ] = AB BA. The accompanying observabes A and B evauated in the state are given, respectivey, as A( q, p ) = ( / )Aˆ and B( q, p ) = ( / )Bˆ according to Definition Having obtained A( q, p ) and B( q, p ), we can evauate the Poisson bracket of A and B in a usua way: A B A B {A,B} =. (3.3.) q i i pi pi qi Lemma 3.3. If an observabe A( q, p ) is stationary in the eigenfunction of Ĥ, then {A, H} = [ AH, ] =0. (3.3.3) Proof: For the given condition, we have Hˆ = E n and Â= A n, which yieds AH = HA = A n E n and hence [ AH, ] = ( AH HA ) =0. On the other hand, evauating A in the state resuts in A( q, p ) = ( / )A ˆ= ( / )An= An. This means that A( q( t), p ( t)) is conservative aong any phase-space trajectory ( q( t), p ( t)) in the state. From the viewpoint of Hamiton mechanics, the conservation of A( q( t), p ( t)) requires d A A A H A H A( q( t), p ( t)) =0= qi pi dt + = q p, q p p q i i i i i i i i where q ( t) and p ( t) satisfy the Hamiton equations (3...9) specified by. The above equation amounts to { AH, } =0 which, together with [ AH=0,, ] gives Eq.(3.3.3). We say that A and H satisfying Eq.(3.3.3) are compatibe with each other. For two operators A and B that are incompatibe, their commutator and Poisson bracket are both nonzero, but usefu reation between { AB, } and [ AB, ] sti exists. One typica exampe comes from the case of ˆx and p ˆx, which have commutator [ xp, x ] = iħ and Poisson bracket { xp, x } =. Expressing them in a form anaogous to Eq.(3.3.3), we have i ħ{ xp, x} = [ xp, x ] = iħ. (3.3.4) This suggests an identity that the operator corresponding to the observabe iħ { AB, } is [ AB, ], i.e., 65

11 i ħ { AB, } = [ AB, ]. (3.3.5) This reation does hod for most commony used quantum operators. The foowing exampes cite two of them. Exampe Given the two observabes A= Lx= ypz zpx and B= Ly= zpx xpz, we have their Poisson bracket as { Lx, Ly } = xpy ypx= Lz ; on the other hand, the commutator of the associated operators L ˆx and L y is known to be [ L, ] i x Ly = ħ Lz. Accordingy, we have [, ] (i L ) i ˆ x Ly= Lz Lz i Lz i { Lx, Ly } ħ = ħ = = ħ ħ, (3.3.6) which satisfies the identity (3.3.5). Exampe Consider A= H and B= x, and their commutator [ Hx, ] = ( i ħ / mp ) x. The observabe reated to [ Hx, ] is found from the reation ˆ iħ i [ Hx, ] px ħ = = px m. (3.3.7) m This observabe is to be inked to the Poisson bracket { Hx, } with H given by Eq.(3..6), H x { Hx, } = = px (3.3.8) p x m x The combination of Eq.(3.3.7) and Eq.(3.3.8) yieds i ħ { Hx, } = [ Hx ˆ, ˆ], (3.3.9) which again is a specia case of Eq.(3.3.5). 3.4 Orbita and Spin Anguar Momentum The above discussions on quantum Hamiton mechanics and the reated quantum probems are a imited to Cartesian coordinates. In this section, we sha continue to demonstrate how quantum operators can be derived directy in spherica coordinates under the framework of Hamiton mechanics. According to the definition of the Quantum Hamitonian H in Eq.(3..6), we first have to express p p and p in spherica coordinates ( rθφ,, ) as pθ pφ p p = pr+ +, (3.4.a) r r sin θ pφ p = ( rpr sinθ) ( pθ sinθ) + + r sinθ r θ φ sinθ (3.4.b) pr pθ pφ = rpr sinθ+ pθ cosθ+ r sinθ + sin θ +. r sinθ r θ sinθ φ Before evauating pr / r, p θ / θ, and p φ / φ, we need to express p r, p θ, and p φ as expicit functions of r, θ, and φ. These expicit expressions are afforded by Eq.(3..8) as S n S n S n pr= = i ħ, pθ= = i ħ, pφ= = iħ, (3.4.) r r θ θ φ φ where ( r, θφ, ) is the given wavefunction. By appying Eq.(3.4.) to the differentiations invoved in Eq.(3.4.b), the quantum Hamitonian H defined in Eq.(3..6) becomes n n H ħ p ħ r p ħ r p ħ = θ+ pθ cotθ + m i r i r mr i i θ 66

12 n L + pφ Vr (, θφ, ) Pr V ħ + = + + sin θ φ. (3.4.3) m mr It can be seen that the Hamitonian H is uniquey determined by the given wavefunction ( r, θφ, ). The foowing theorem gives the dynamic representation of in spherica coordinates. Theorem 3.4. (Yang, 006D) A quantum state ( r, θφ, ), expressed in spherica coordinates, is a dynamic system whose behavior obeys the foowing Hamiton equations: n n cot n r = ħ + ħ, θ = ħ + ħ θ, φ = ħ (3.4.4) im r imr imr θ imr imr sin θ φ Proof: The dynamic representation of the state in spherica coordinates can be derived from Eq.(3..9a) with H given by Eq.(3.4.3): H pr H p cot H p r ħ θ θ φ, θ ħ = = + = = +, φ = =. (3.4.5) p m imr p mr imr p mr sin θ r θ On deriving the above Hamiton equations, the canonica variabes q = ( rθφ,, ) and p = ( pr, pθ, pφ) must be regarded as independent variabes in the quantum Hamitonian H, as in cassica Hamiton mechanics. The adjoin Hamiton equations ṗ= H / q from Eq.(3..9b) are redundant, since, according to Theorem 3.., the soution for p is aready given by Eq.(3.4.), as ong as is a soution of the Schrodinger equation. By substituting Eq.(3.4.) into Eqs.(3.4.5), we obtain the dynamic representation of as in Eqs.(3.4.4). In comparison with their cassica counterparts, the mechanica momenta Pr and L in quantum Hamitonian (3.4.3) contain additiona quantum correction terms: ħ n Pr p ħ = r+ p r+ i r i r, (3.4.6a) ħ n Lz n L p ħ = θ+ pθ cot θ+ +, Lz= p φ i i θ ħ, (3.4.6b) sin θ φ where the terms invoving Panck constant stem from quantum correction. The corresponding operators for Pr, L, and L z can be identified by expressing Eqs.(3.4.6) in the form of Eq.(3.3.). Theorem 3.4. (Yang, 006D) The momentum operators P ˆr, L, and L z corresponding to the observabes P r, L, and L z defined in Eqs.(3.4.6) are given by ħ P, cot, r L θ ħ = L z i + r r = ħ + + = θ θ sin θ φ, (3.4.7) i φ in terms of which the spherica Hamitonian operator can be expressed as H P = r+ L + V. (3.4.8) m mr Proof: Substituting pr= S / r= i ħ (n )/ r from Eq.(3.4.) into Eq.(3.4.6a) yieds ħ P r ħ ħ = + = + + r r r i r r i r r. Comparing the above equation to the definition P ( / ) ˆ r = Pr gives the desired expression for P ˆr, the associated operator for the inear momentum in the r direction. Simiary, The expression of Eq.(3.4.6b) in terms of the wavefunction by using p θ = i ħ (n )/ θ and p φ = i ħ (n )/ φ gives rise to ħ L cot θ, L z ħ ħ = + + = θ θ sin θ φ i φ i φ, (3.4.9) from which the definitions of L = ( / ) Lˆ and L = ( / ) Lˆ yied the expressions for ˆL, and 67 z z φ

13 L ˆz. We thus have obtained the operators P ˆr, ˆL, and L ˆz directy from their associated observabes defined in quantum Hamiton mechanics without using quantization axiom. In the same way, we can obtain the Hamitonian operator Ĥ by writing Eq.(3..4.3) in the form of H= ( / ) Hˆ as ħ H= + + cotθ V m r r r r θ θ sin θ φ (3.4.0) P = r + L V + m mr from which Ĥ reads as in Eq.(3.4.8). It seems that Ĥ may be obtained by simpy appying the quantization axiom to the cassica Hamitonian H= pr / m+ L /( mr) + V, but it is noted that p r shoud be repaced by the mechanica momentum operator P ˆr in Eq.(3.4.7) instead of the canonica momentum operator pˆ r= i ħ / r. This situation is what we had encountered in standard quantum mechanics that the quantization axiom p iħ cannot be appied directy to Spherica coordinates. In case of centra-force fied V= Vr ( ), we have [ L, H ] = [ L, z H ] =0, which impies that the observabes L and L z are stationary in an eigenstate of Ĥ. From Lemma.3., it means that L and L z are conservative aong any phase-space trajectory ( q( t), p ( t)) determined from Eqs.(3.4.4). Since we have aready derived the expressions for L and L z, we can verify their conservation by simpy showing dl / dt= dlz / dt=0. With the property that the wavefunction has a separabe soution ( r, θφ, ) = Rr ( ) Θ( θ) Φ ( φ) for centra-force probem and with the expressions for H, L and L z in Eq.(3.4.3) and Eq.(3.4.6b), we obtain readiy the expected resuts dl z {, } L z H L z H z, dl L H { L, H} L H L = = =0 = = H =0 dt φ p p φ dt θ p p θ φ φ θ θ As Lemma 3.3. indicates, the conserved vaues for H, L and L z in the state are just the eigenvaues of Ĥ, ˆL, and L ˆz, respectivey. This property is refected in the reations of ˆL Ĥ= H, = L, and ˆz L= Lz, as expressed in Eq.(3.4.0) and Eq.(3.4.9). Given a quantum state ( r, θφ, ), we determine its dynamica representation from Eqs.(3.4.4); on the other hand, aso gives us the probabiity of the partice s spatia distribution. Consequenty, there exists cose reation between these two different descriptions of. Theorem The stabe equiibrium radia position predicated from the dynamic representation of in Eqs.(3.4.4) is identica to the position with the maximum radia probabiity Pr ( ) =4 πrr ( rrr ) ( ) determined from ( r, θφ, ) = Rr ( ) Θ( θ) Φ ( φ) ; or stated mathematicay, dr d =0 Pr ( ) =0. (3.4.) dt dr Proof: Using ( r, θφ, ) = Rr ( ) Θ( θ) Φ ( φ), we may rewrite Eq.(3.4.4a) as d r = ħ n ( rrr ( )). (3.4.) imdr Hence, the equiibrium position for r is found from the condition r = d(n( rr))/ dr=0, i.e., drr ( )/ dr=0. On the other hand, the radia probabiity function has the expression Pr ( ) = 4 πrr ( rrr ) ( ) =4 π( rrr ( )) by noting that Rr ( ) is a rea function of r for centra- force probems. Accordingy, the maximum radia probabiity occurs at the ocation of dp / dr= drr ( )/ dr=0, which is just the condition of r=0. Exampe 3.4. (Yang, 005A) r / a0 For hydrogen atom at ground state, we have Rr ( ) = e with a0=4πε0ħ /( me ) being the Bohr radius and Θ ( θ) =Φ ( φ) =. Substituting this wavefunction into Eqs.(3.4.4) yieds the equations of motion for the ground-state eectron as: 68

14 69 dρ ρ dzθ zθ dφ =, = i, =0, zθ= cosθ (3.4.3) dτ i ρ dτ ρ dτ where ρ= r / a0 is the dimensioness radia distance and τ= tħ /( ma0) is the dimensioness time. Eq.(3.4.3) shows that the ground-state hydrogen atom has an equiibrium radia position r eq at the Bohr radius a 0, i.e., ρ=. Meanwhie, the radia probabiity function is given by Pr ( ) =4 πrrr ( ) r / a0 =4 πre that yieds a maximum vaue at r max = a 0 by etting dpr ( )/ dr=0. Therefore, we have r eq = r max = a 0. It is not surprising that the probabiity of finding a partice at the stabe equiibrium position has a maximum vaue, since there is aways a restoring force acting on the partice toward the stabe equiibrium position, and once the partice reaches the stabe equiibrium position, it wi remain there as ong as no disturbance is appied. However, Eq.(3.4.) aone cannot te us directy the stabiity of the equiibrium points. The information of the force action around the equiibrium points is required to judge their stabiity. The force information is provided by the tota potentia VTota = Q( ( q )) + V with Q ( ( q )) being the quantum potentia determined from Eq.(3..7). With the hep of Eq.(3.4.4), the Hamitonian in Eq.(3.4.3) can be recast into the foowing form m H= r ( rθ) ( rφ sin θ) + + VTota + ( r, θφ, ) = E, (3.4.4) where the tota potentia is expressed in terms of the wavefunction as ħ ħ n n n VTota= ( 4+ cot θ) Vr ( ) mr m r r θ r sin θ φ. (3.4.5) When appied to atomic modes, the tota potentia V Tota exhibits the observed atomic she structure. Exampe 3.4. (Yang, 005A, 006C) Considering hydrogen atom, we have wave function nm ( r, θφ, ) = R ( ) ( ) ( ) nrθm θφ m φ, where Rn ( r ) is expressed in terms of the Laguerre poynomia L β αρ ( ) as Rn ( ρ ) = ρ/ n + ( ρ / n) e Ln ( ρ / n), ρ= r / a0, n N; Θ m ( θ) is expressed in terms of the associated β m im Legendre poynomia Pα ( z) as Θ m ( θ) = P (cos ) θ, =0,,,, n, and Φ m ( φ) = e φ, m =0, ±, ±,, ±. The tota potentia V Tota is state-dependent, and the V Tota reating to nm is denoted by V nm as d n Rn ( ρ) d n m ( θ) Vnm VTota / ħ Θ = ( cot θ ) ma = + 4+, (3.4.6) 0 ρ 4ρ dρ ρ dθ where the first term V= / ρ is the dimensioness Couomb potentia and the remaining terms in V nm constitute the quantum potentia Q. The she structure observed in hydrogen atom is actuay caused by V nm, and the quantum force derived from V nm provides the necessary driving force to maintain the eectron within the she. Two typica exampes V300 and V30 are considered here, which are given by V V cot θ 4 ( 4ρ 8) ( ρθ, ) = + +, (3.4.7) ρ 4ρ ρ 8 ρ+7 ( ρ 8 ρ+7) +4 tan θ+ cot θ ( ρθ, ) = + + ρ 4ρ ( ρ 6 ). (3.4.8) V300 has three ayers distributed in the radia direction and V30 has four ayers with two of them in the radia direction and the other two in the azimuth direction, as shown in Fig.3.4. and Fig.3.4. According to the definition of V nm in Eq.(3.4.6), the radia ayers are separated by the infinite potentia barriers ocated at the points: ρ/ n + ρ=0, and R ( ρ) = ( ρ / n) e L ( ρ/ n) =0. (3.4.9) n For a given vaue of n, there are n different vaues of ρ satisfying Eq.(3.4.9), and hence at most n shes can be divided by these vaues of ρ, when =0. In the case of n=3, =0, n

15 Eq.(3.4.9) gives ρ( ρ 8 ρ+7 ) =0, which can aso be obtained directy from the denominator of V300 in Eq.(3.4.7). The three ayers of V300 are then ocated, respectivey, in the ranges of 0< ρ< ( 9 3 3)/, ( 9 3 3)/ < ρ< ( 9+3 3)/, and ρ> ( 9+3 3)/, as is shown in Fig.3.4. for a 3D surface pot. From probabiity consideration, the probabiity that the eectron appears at the boundary of the ayer is zero, since the radia probabiity density Pn ( ρ) =4 πρ Rn ( ρ) is identica to zero at the ayer boundary determined from Eq.(3.4.9). The she structure of V30 is distributed in the both radia and azimuth directions. The number and the separation of the azimuth shes are determined by the roots of Θ m ( z ) θ. From Eq.(3.4.6), the potentia approaches infinity at the foowing azimuth ange: V Tota ( ρ, θ ) Third She First She Second She x z Fig.3.4. The tripe-she structure of the tota potentia VTota ( ρθ, ) for the quantum state n=3, = m =0, showing that the potentia approaches infinity at the she boundaries where the eectron cannot reach and the probabiity density function Pn ( ρ) =4 πρ Rn ( ρ) is competey zero. The she boundary ocates at the root of ρ( ρ 8 ρ+7 ) =0, and three ayers are then formed, respectivey, in the ranges of 0< ρ< ( 9 3 3)/, ( 9 3 3)/ < ρ< ( 9+3 3)/, and ρ> ( 9+3 3)/, m m / d θ m θ θ m θ dzθ 70 z =, Θ ( z ) = ( z ) Pz ( ) =0, (3.4.0) where z θ = cosθ. For a given quantum number, the number of azimuth shes is m +, and the maximum number is + in the case of m =0. Accordingy, V30 has two azimuth shes with boundaries at z θ = and Θ 30( zθ) = zθ=0, i.e., at θ=0 and θ= π /. On the other hand, the number of radia shes of V30 is equa to n = with inner she in the range of 0< ρ<6 and outer she in the range of ρ>6, as shown in Fig Combining together the radia and azimuth shes, we have totay four shes with each containing one equiibrium point: () first inner she occupying the region of 0< ρ<6, and 0< θ< π/ with ( ρeq, θeq) = ( 3,cos / 3 ), () second inner she occupying the region of 0< ρ<6, and π / < θ< π with ( ρeq, θ eq) = ( 3, π cos / 3 ), (3) first outer she occupying the region of ρ>6, and 0< θ< π/ with ( ρeq, θ eq) = (,cos / 3 ), and (4) second outer she occupying the region of ρ>6, and π / < θ< π with ( ρeq, θ eq) = (, π cos / 3 ). A three-dimensiona iustration of the four shes of V30 is shown in Fig By evauating the radia probabiity density function P ( ρ) =4 πρ R ( ρ), it can be checked that the maximum vaue of P ( ρ ) just occurs at the n n n

16 equiibrium points ( ρeq, θ eq), i.e., the owermost point in each subshe, as proved in Theorem With given V Tota, the quantum forces in the three directions can be determined, respectivey, as fr = VTota / r, fθ= VTota / θ and fφ= VTota / φ. These quantum forces together with the Couomb force estabish the stabe configurations for atomic structure, as wi be discussed in Chapter 5. V Tota ( ρ, θ ) First inner she Second inner she First outer she x z Second outer she Fig.3.4. The quadrupe-ayer structure of the tota potentia V30 ( ρθ, ), showing that the inner and outer ayers are respectivey divided into two sub-ayers aong the azimuth direction with ranges of 0< θ< π/ and π/ < θ< π. Totay four shes are formed with each containing one equiibrium point: () first inner she occupying the region of 0< ρ<6, and 0< θ< π/ with ( ρeq, θeq ) = ( 3,cos / 3 ), () second inner she occupying the region of 0< ρ<6, and π/ < θ< π with ( ρeq, θ eq) = ( 3, π cos / 3 ), (3) first outer she occupying the region of ρ>6, and 0< θ< π/ with ( ρeq, θeq) = (,cos / 3 ), and (4) second outer she occupying the region of ρ>6, and π/ < θ< π with ( ρeq, θ eq) = (, π cos / 3 ). It is we known that the anguar momentum L defined in Eq.(3.4.9) invoves ony orbit motion. By contrast, the anguar momentum Pθ = mr θ derived in Eq.(3.4.5b) contains both spin and orbit motions. The corresponding oca anguar momentum P θ can be found from the reation (3.4.5b) or Eq.(3.4.4b) as cotθ Pθ = mr θ ħ ħ = + = Lθ+ S, (3.4.) i θ i where L θ is the θ -component orbita anguar momentum, and S is the oca spin anguar momentum. It can be seen that the oca spin S= ħ /( i)cotθ is independent of the wavefunction and thus aso independent of the appied potentia V. Like the proof of the quantization of action variabe mentioned in Theorem 3.3., the quantization of the mean vaue of L θ can be determined from the contour integra over a cosed path c θ in the θ compex pane as 7

17 ħ ħ dθ ħ Lθ = Ld θ θ= dθ= dθ= d(n θ) π cθ πi cθ θ πi cθ dθ πi c = ħn, n =0,,, 3, θ θ (3.4.) where θ( θ ) is a function of the singe variabe θ obtained from ( r, θφ, ) by treating r and φ as constants. The function θ maps the cosed path c θ in the compex θ pane into a cosed path c in the compex θ pane, and the quantum number n θ is the number of encircement of the origin in the θ pane by the cosed path c. Anaogousy, we have the quantization of the φ -component orbita anguar momentum as ħ L φ = d φ = n φ, n φ =0,,, 3, π ħ. (3.4.3) c φ i θ We now turn the attention to the spin quantization. Using Eq.(3.4.), the mean oca spin is given by ħ S = Sdθ= cot θdθ π c 4πi. (3.4.4) c θ Noting the residue of cotθ at its poes θ= nπ, n Z, is equa to, we have from the residue theory S θ ħ ħ = ( πinθ) = ns, ns=0,,, 3,, (3.4.5) 4πi where n s is the number of zero of sinθ, i.e., nπ, n Z, encosed by the contour c θ, Im( θ ) and aso note that since sinθ is an anaytica function, the C 3 number of poe of sinθ C C C 0 encosed by c θ is aways zero. It is worth noting that upon arriving at the resut (3.4.5), π π π π 3 π Re( θ ) we do not specify the type of partices, neither the type of the appied potentia. The spin Fig The quantization of spin is characterized by the number of quantization rue (3.4.5) says nπ, n Z, encosed by the θ compex trajectories. In the figure, the that the vaue of the mean spin contour C 0 encoses no point of nπ, indicating that the partice is ony aowed to be integer tracing the contour C 0 is a spiness partice, whie the contour C encoses one point of nπ, indicating that the partice tracing C mutipe of ħ / ; furthermore, is a spin-/ partice. Simiary, the partices tracing C it provides us with a and C 3 are spin- and spin-3/ partices, respectivey. geometrica method to identify the spin of a given partice by inspecting any of its θ trajectory c θ and counting the number of the point nπ, n Z within it. The θ trajectory can be found by integrating Eqs.(3.4.4). Referring to the demonstration in Fig.3.4.3, the contour C 0 encoses no point of nπ, indicating that the partice tracing the contour C 0 is spiness, whie the contour C encoses one point of nπ, indicating that the partice tracing C is a spin-/ partice. Simiary, the partices tracing C and C 3 are spin- and spin-3/ partices, respectivey. The practica computation of the eectron trajectory in the hydrogen atom using the Hamiton equations of motion (3.4.4) shows that the reated trajectory beongs to the type of C and thus confirms an eectron as being a spin-/ partice. Corresponding to the canonica momenta p r, p θ, and p φ, we can find their associated canonica operators p r, p θ, and p φ from Eq.(3.4.) : ħ ħ ħ pr=, pθ=, pφ=, (3.4.6) i r i θ i φ and from the definition (3.3.) 7

18 73 pr= p r, pθ= p θ, pφ= pφ. (3.4.7) The comparison of Eq.(3.4.6) with Eq.(3.4.7) gives ħ ħ ħ p r=, p θ=, pφ= (3.4.8) i r i θ i φ The canonica momenta p r, p θ, and p φ must not be confused with the mechanica momenta Pr = r, Pθ = mr θ, and P mr φ φ sin = θ, which can be found from Eqs.(3.4.4) as P r mr ħ P r i r r = = + = (3.4.9a) cotθ P mr P θ= θ ħ = + = θ i θ (3.4.9b) P mr sin P φ= θ θ ħ = = φ i φ (3.4.9c) from the which the mechanica momentum operators can be identified as ħ cotθ P,, r ħ Pθ ħ = Pφ i + = + = r r i θ. (3.4.30) i φ It can be seen that unike Cartesian coordinates, in spherica coordinates the canonica momentum operators p r, p θ, and p φ are distinct from the mechanica momentum operators P r, Pθ, and Pφ in the additiona terms ħ i /r and ( i ħ / )cotθ. It can be seen that P ˆr and P φ= L z are identica to those derived in Eq.(3.4.7), where they have been obtained aternativey from the expressions for P r and p φ. Here we obtain the same P ˆr and ˆP φ directy from the Hamiton equations of motion (3.4.4). The combination of ˆP θ and ˆP φ in Eq.(3.4.30) affords us the information of tota anguar momentum. Theorem In spherica coordinates, the tota anguar momentum operator is given by Pφ ( / ) J = Pθ+ = L + ( ħ / ) + ħ, (3.4.3) sin θ sin θ where L ˆ is the usua orbit anguar momentum operator defined in Eq.(6.7). Proof: The combination of the θ -component and the φ -component anguar momenta gives the tota anguar momentum vector as P φ J= P P L z θeθ+ eφ= θeθ+ e φ, (3.4.3) sinθ sinθ which has the operator representation: Lz J= Pθeθ+ e φ, (3.4.33) sinθ where e θ and e φ are, respectivey, the unit vectors in the θ and φ directions. Using the definitions of ˆP θ and P φ= L z from Eq.(3.4.30), the operation represented by J ˆ becomes where ˆP θ and ˆP φ are evauated, respectivey, as ( ) P J= J J = P θ+ φ, (3.4.34) sin θ ħ cotθ cotθ Pθ= Pθ( Pθ) = + + i θ θ ( / ) cot θ ħ = ħ + + ( / ) + θ θ ħ sin θ (3.4.35a)

19 P ˆ ħ ħ = φ = i φ i φ ħ. (3.4.35b) φ The substitution of ˆP θ and ˆP φ into Eq.(3.4.34) gives rise to the representation of Ĵ as ˆ ( / ) cot θ ħ J = ħ ( / ) + θ θ sin θ φ ħ, (3.4.36) sin θ where the first three terms in J ˆ is the orbita anguar momentum operator L ˆ aready derived in Eq.(3.4.7) and the remaining two terms are due to spin motion 3.5 Compex Hamitonian and Compex Energy Since in compex mechanics, position and momentum are defined in a compex domain, it is natura to expect that energy in compex mechanics is aso defined in a compex sense. Let us begin this discussion with the harmonic osciator in Exampe 3... Cassica Hamitonian for a harmonic osciator in dimensioness form is given by and its reated Hamitonian operator reads H ( c) = ( p+ x), (3.5.) ħ H = p + x= + x. (3.5.) x 74 Soving the eigenvaue probem of H = E, we can find the eigenvaue and eigenfunction as x /, n n n, n = CH ( x ) e, E = n+/, n=0,,,. (3.5.3) It can be seen that the eigen energies E,n of a harmonic osciator are rea. Rea energy has a deeper impication in compex mechanics, if we ca that the position and momentum are both compex and consider why energy composed from them is rea. According to Eq.(3..8), the equation of motion in the eigen state,n is given by dx dn( x) = p= ħ. (3.5.4a) dt i ( x ) dx Especiay for n=0 and n=, we have n dx n=0 : = ix, (3.5.4b) dt dx x n= : = i. (3.5.4c) dt x We can see that the momentum p given by Eq.(3.5.4a) is compex and the position x ( t) soved from Eq.(3.5.4a) is aso compex. However, the tota energy formed from p and x ( t) are rea, as shown in Eq.(3..4): d n n( x), n dx H ( x, p ) = p + x = E = n+/, (3.5.5) where p C and x C, but E n is rea. Such a situation that compex position and momentum resut in rea energy is not an accident. Not every Hamitonian operator H has rea eigenvaues. ( c) ( c) In conventiona quantum mechanics, one imposes the Hermitian condition H = H, where represents compex conjugate transpose, to ensure that the Hamitonian operator H has a rea eigenvaue (rea energy). It was pointed out by Bender [998] that the reaity of the eigenvaue of H is actuay due to a weaker condition caed PT symmetry. PT represents combined parity refection P and time reversa T. The effect of parity refection P is to make spatia refections, p p and x x, whie the effect of time reversa T is to make the repacements p p, x x, and i i. Consider the foowing cassica Hamitonian