Applying classical geometry intuition to quantum spin

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1 European Journal o Physics PAPER OPEN ACCESS Applying classical geometry intuition to quantum spin To cite this article: Dallin S Duree and James L Archibald 06 Eur. J. Phys View the article online or updates and enhancements. Related content - Quantum Mechanics: Angular momentum M Saleem - Corrigendum: A ull quantum analysis o the Stern Gerlach experiment using the evolution operator method: analysing current issues in teaching quantum mechanics E Benítez Rodríguez, L M Arévalo Aguilar and E Piceno Martínez - An operational approach to spacetime symmetries: Lorentz transormations rom quantum communication Philipp A Höhn and Markus P Müller This content was downloaded rom IP address on 3/08/08 at :3

2 European Journal o Physics Eur. J. Phys. 37 (06) (9pp) doi:0.088/ /37/5/ Applying classical geometry intuition to quantum spin Dallin S Duree and James L Archibald Department o Physics and Astronomy, Brigham Young University, Provo, UT 8460, USA dallin_duree@byu.edu Received April 06, revised 0 June 06 Accepted or publication 0 June 06 Published 0 July 06 Abstract Using concepts o geometric orthogonality and linear independence, we logically deduce the orm o the Pauli spin matrices and the relationships between the three spatially orthogonal basis sets o the spin-/ system. Rather than a mathematically rigorous derivation, the relationships are ound by orcing expectation values o the dierent basis states to have the properties we expect o a classical, geometric coordinate system. The process highlights the correspondence o quantum angular momentum with classical notions o geometric orthogonality, even or the inherently non-classical spin-/ system. In the process, dierences in and connections between geometrical space and Hilbert space are illustrated. Keywords: spin, quantum mechanics, orthogonality (Some igures may appear in colour only in the online journal). Introduction It is possible to ind the Pauli spin matrices and the corresponding geometrically orthogonal basis sets or the spin-/ system using arguments rom classical geometry. Like the vector model [, ], this undertaking leverages classical understanding to build intuition or quantum angular momentum. But rather than using a semi-classical model, our approach uses completely quantum spin states. It involves the application o classical principles to expectation Currently at AMS-TAOS USA Inc. Original content rom this work may be used under the terms o the Creative Commons Attribution 3.0 licence. Any urther distribution o this work must maintain attribution to the author(s) and the title o the work, journal citation and DOI /6/ $ IOP Publishing Ltd Printed in the UK

3 Eur. J. Phys. 37 (06) D S Duree and J L Archibald values, in a manner similar to Ehrenest s theorem [3, 4]. The technique can provide valuable insight or anyone with an advanced undergraduate or introductory graduate understanding o quantum mechanics. This method illuminates the correspondence between Hilbert space, which is used to mathematically describe quantum states, and the geometrical space we use in classical descriptions o angular momentum. It makes connections between the parameters used to deine a Cartesian coordinate system and the ree parameters selected when deining geometrically orthogonal sets o basis states to describe a quantum spin-/ particle. The procedure clariies some common misconceptions about the quantum picture o angular momentum, in particular the conusion students oten have distinguishing between geometric and Hilbert space, discussed in [5]. It does not explain the why o quantum spin, but ocuses on how we work with quantum angular momentum states. There are several rigorous ways to derive the well-known relationships between the x, y, and z basis sets o the spin-/ system and the corresponding Pauli matrices. These include the use o rotation operations [6, 7], ladder operators [8], direct diagonalization [9], or symmetry arguments [0, ]. Here we deduce, rather than derive, the relationships. We limit our discussion to a spin-/ system or simplicity and clarity. This also avoids dealing with dierent types o unbiased states []. For example, as discussed later, or a spin-/ particle described using the z basis, the only states with a zero expectation value o the z component o angular momentum are ones made o an equal superposition o spin up and spin down. For a spin- particle, this is not only achieved with an equal superposition o m =-and m =, but also with the m = 0 basis state, or any combination o all states or which the and basis states have amplitudes o equal size. The canonical spin-/ system illustrates the universality o the core principle behind Ehrenest s theorem, even in systems with no classical analogue [3, 4]; even though you cannot explain an electron s intrinsic angular momentum as the result o physical rotation in the classical sense, the expectation values o a spin-/ system do obey principles o classical geometry. While the origin o spin cannot be described semi-classically, classical intuition can be applied to better understand the consequences o spin.. Quantum spin-/ ormalism It is common to describe spin-/ particles using a basis consisting o the two eigenstates o the z-component o angular momentum, z and z. In vector notation, we can write these states as () () = and = 0 z z 0. ( ) We will use the z basis as our starting point to deduce the orm o the Pauli spin matrices and ind the relationships between the x, y, and z basis states. We assume that the z basis exists and that every possible state o our particle can be expressed as a sum o these two states in the orm y = a a + b = b, ( ) z z () where a and b are scalar, potentially complex, constants. In matrix notation, the operators that yield inormation about the x, y, and z components o spin can be written as times the Pauli spin matrices. The Pauli spin matrix s z can be easily ound by noting that it is the matrix or which and z are the eigenvectors with z

4 Eur. J. Phys. 37 (06) D S Duree and J L Archibald eigenvalues o + and. This gives us the matrix s = z ( ) ( ) I we measure the z-component o angular momentum or a spin-/ particle in an arbitrary state, we will always get either plus or minus, with probabilities o a* a and b* b i the state is normalised. To predict the outcome o a measurement along a dierent axis, we can write our quantum state in terms o the eigenstates o the component o angular momentum we plan to measure. Since the z axis is chosen arbitrarily, we know that a pair o eigenstates o any component o angular momentum along any axis must exist and have similar properties to the z basis states. Two special basis sets are made up o the eigenstates o angular momentum along the x and y axes, respectively. Because the z basis orms a complete set, we should be able to write these basis states in terms o the z states. We will start by writing the x basis states as x = A z + B z and ( 4) = C + D, ( 5) x z z where A, B, C, and D are constants. We can deduce what these constants must be simply by considering what properties these states should have. In the process, we will see connections between physical (geometric) space and Hilbert space. 3. Orthogonality To ind A and B in equation (4) we note that a classical particle with its angular momentum in the x direction will have no component o angular momentum in the z direction. For a quantum spin-/ particle we will always measure the z component to be, never zero, regardless o the particle s state. So rather than mapping our intuition o geometrical orthogonality onto possible measurement outcomes, we will make our x basis states geometrically orthogonal to z by setting the expectation value o the z component o angular momentum to zero. For the expectation value to be zero, it must be exactly as likely to measure the z component o spin to be - as it is to measure +, such that the two possibilities cancel each other out. As such, we intuitively expect that x should be an equal superposition o z and z. To show this more rigorously, we use the operator S z = ( z z - z z ) = s z ( 6) to ind the expectation value o the z component o angular momentum. Using this operator, we get x Sz x = ( A* A - B* B). ( 7) We require that this expectation value be zero, such that AA * - BB * = A - B = 0. ( 8) And, likewise, since the x and y basis sets will be complete, we should be able to write the z basis states in terms o the x or y basis states. 3

5 Eur. J. Phys. 37 (06) D S Duree and J L Archibald As we expected, A and B must have the same magnitude, giving an equal superposition o z and z. But the complex phases o A and B are unrestricted. So the most general orm or x, subject only to the limitation that it be normalised and geometrically orthogonal to the z basis states, is = + x [ e i z e i z ], ( 9 ) where and are arbitrary real constants. It makes sense that we get one arbitrary phase angle, since quantum mechanics always allows us an arbitrary overall phase actor. But why two? There s a geometric explanation or this reedom. Ater deining the z axis o a Cartesian coordinate system, the x axis can point in any one o an ininite number o directions which are orthogonal to z. These choices can be parametrised by an angle relative to some reerence direction, as shown in igure. We will set and to zero both or simplicity and because it gives us the conventional orm o the basis state x = [ z + z ]. ( 0) 4. Linear independence To ind x we note that it, too, must be geometrically orthogonal to the z basis. So it must have the orm = + x [ e i 3 z e i 4 z ]. ( ) We ind an additional constraint when we consider that the x basis states could have been the z basis states had we simply chosen a dierent direction or our z axis. As such, since our z basis states are orthogonal to each other, we expect the two states in the x basis to be orthogonal to each other as well. As is oten done, we have unortunately used the word orthogonal to mean two dierent things. When we say that the x states must be orthogonal to the z states, we are reerring to geometric orthogonality. But when we say that the x states must be orthogonal to each other, we reer to orthogonality in Hilbert space or linear independence. Just as a dot product o zero assures geometric orthogonality, an inner product x x = 0 guarantees that x and x are linearly independent, such that one cannot be written in terms o the other. The inner product o the x basis states is = + j + j x x [ z z ][ ei 3 z ei 4 z ] ( ) = ( j + j e i 3 e i 4). For this to be zero, the two phases must dier by π radians. I we choose the arbitrary global quantum phase o this state such that 3 = 0, both or simplicity and by convention, we ind that ei 4 =-and x = [ z - z ]. ( 3) 4

6 Eur. J. Phys. 37 (06) D S Duree and J L Archibald Figure. Once the z axis o a Cartesian coordinate system is selected, there are still an ininite number o possible choices or the x axis which are all orthogonal to z. These choices are parametrised by the variable θ in the igure. Knowing the orm o x and x, it is simple to show that the Pauli spin matrix s x is given by s = 0 x ( 0 ). ( 4 ) 5. The y basis Just as we saw or the x basis states, or the y basis states to be normalised and geometrically orthogonal to the z axis, they must have the orm = + y [ e i 5 z e i 6 z ], ( 5 ) = + y [ e i 7 z e i 8 z ]. ( 6 ) Since we have the reedom to multiply each state by an arbitrary overall phase actor, or simplicity (and to arrive at the canonical orm o the states), we can set 5 and 7 to zero. The other phase angles are constrained by that act that, in addition to being geometrically orthogonal to z, these states must also be geometrically orthogonal to the x axis we have deined. The operator which gives us inormation about the x component o spin can be ound by noting that, when written in the x basis, this operator should look similar to S z represented in the z basis: Sx = ( x x - x x ). ( 7) 5

7 Eur. J. Phys. 37 (06) D S Duree and J L Archibald Figure. Once the x and z axes are selected, there are still two possible choices or the y direction. One choice, labelled y lh in the igure, will result in a let-handed coordinate system. The other choice results in a right-handed coordinate system. To write this in the z basis we plug in equations (0) and (3) to get S x = ( z z + z z ) = s x. ( 8) We can use S x to ind the expectation value o the x component o angular momentum or a particle in the state : y = + - y Sx y ( = 4 e i 6 e i 6) cos ( 6). ( 9) I we want y to be geometrically orthogonal to the x states then this must be zero, implying that 6 =p and giving us only two unique possibilities: y = [ z i z ]. ( 0) As we discuss in the next section, the choice o whether to use the upper or lower sign is not arbitrary, so we will not select one over the other just yet. Applying the same condition to y and orcing the inner product y y to be zero gives us y = [ z i z ]. ( ) 6. Coordinate system handedness There is a geometric explanation or the two possible choices in equations (0) and (). In a Cartesian coordinate system, once the z and x axes have been selected, there are still two 6

8 Eur. J. Phys. 37 (06) D S Duree and J L Archibald possible directions or y. As illustrated in igure, one direction results in a right-handed and the other in a let-handed coordinate system. We can ind the handedness o a geometric coordinate system with cross products. For example, i nˆ j is a unit vector along the jth axis, then or a right-handed coordinate system using a right-handed cross product, nˆx nˆy = nˆ z. For a let-handed coordinate system we get a minus sign: nˆx nˆy = -nˆ z. Similarly, we can ind the handedness o the Pauli spin matrices by noting that the Pauli matrices, multiplied by a constant and combined with the ( ) identity matrix, orm the basis o the SU() Lie group [5]. Then we can evaluate Lie brackets, which are analogous to cross products. I we evaluate the Lie bracket o sz with sx, we get s s = s + = - s -, i i z x y y, ( ) where s y + /s y- is the Pauli spin matrix we get i we choose the upper/lower sign or equations (0) and (). The change in sign in equation () is just what we would expect when going rom a right- to a let-handed coordinate system. Similarly, there is a link between the Pauli matrices and quaternions [6 8], and between cross products and commutators in quaternion algebra [9]. This suggests a connection between cross products in geometric space and commutators in Hilbert space. Because there is no spatial representation o the spin-/ particle s angular momentum operator, we will make an analogy with the orbital angular momentum operator [0]: L = r p. ( 3) Here r is the position and p the momentum operator. Assuming a right-handed coordinate system, the components o L are Lx = ypz - zp y, ( 4) Ly = zpx - xp z, ( 5) Lz = xpy - yp x. ( 6) We can use these components to calculate the commutator o L x with L y. Noting that position and linear momentum operators commute with operators or orthogonal spatial dimensions and that [ z, pz ] = i, it is easy to show that [ Lx, Ly] = ilz. I we had chosen a let-handed coordinate system (but still used a right-handed cross product), we would have gotten the same result but with a minus sign. By analogy, we may suppose that a right-handed coordinate system or a spin-/ particle is the one that results in the commutation relation [ S, S ] = i S, ( 7) x y z while a let-handed coordinate system would result in a minus sign in the commutation relation. We get the commutation relation in equation (7) i we choose the upper sign in equations (0) and (): y = [ z + i z ], ( 8) y = [ z - i z ]. ( 9) 7

9 Eur. J. Phys. 37 (06) D S Duree and J L Archibald From these, the Pauli matrix s y can be ound: s = 0 - i i y ( ) ( ) This is the last piece o the puzzle, and we have now deduced the relationships between the x, y, and z basis states as well as the Pauli matrices or the spin-/ system. 7. Conclusion We deduced the Pauli matrices and the relationships between the x, y, and z basis states or a spin-/ particle using the concepts o geometric orthogonality and linear independence. By insisting that the two states in the x basis be normalized, geometrically orthogonal to the z states, and orthogonal to each other in Hilbert space (linearly independent), we arrived at expressions which were completely speciied except or three arbitrary phase angles two due to arbitrary overall phase actors, and a third related to choosing the direction or the x axis or a given selection o z axis direction in a Cartesian coordinate system. With the y basis we had less reedom because the states had to be geometrically orthogonal to both the z and the x basis states. We again had an arbitrary overall phase actor or each basis state. But we only had two possible choices or the remaining phase actors, similar to the choice o handedness in a Cartesian coordinate system. We determined handedness by making a connection between cross products and commutators. This intuitive exercise illustrates the connection between classical geometric space and quantum Hilbert space, even or spin-/ systems, which are intrinsically not classical. Acknowledgments We thankully acknowledge Jean-Franois Van Huele or very helpul discussions, and we are grateul to Christopher J Erickson and Manuel Berrondo or eedback on this manuscript. This research was unded by NSF Grant PHY and by BYU s College o Physical and Mathematical Sciences. Reerences [] Beiser A 995 Total angular momentum Concepts o Modern Physics 5th edn (New York: McGraw-Hill) ch 7.8, pp [] Loh Y L and Kim M 05 Am. J. Phys [3] Ehrenest P 97 Z. Phys [4] Thompson W J 004 Angular momentum in quantum systems Angular Momentum (Chapel Hill, NC: Wiley-VCH) ch 5, pp 69 0 [5] Zhu G and Singh C 0 Am. J. Phys [6] Merzbacher E 998 Quantum Mechanics 3rd edn (New York: Wiley) ch 6, pp 38 6 [7] Sakurai J J 994 Modern Quantum Mechanics (Revised Edition) (Reading, MA: Addison Wesley Longman) ch 3, pp [8] Griiths D J 005 Introduction to Quantum Mechanics nd edn (Englewood Clis, NJ: Pearson Prentice Hall) ch 4, pp [9] Narducci L M and Orszag M 97 Am. J. Phys [0] Dirac P A M 958 The spin o the electron The Principles o Quantum Mechanics 4th edn (Oxord: Clarendon) ch 37, pp 49 5 [] Epstein S T 966 Am. J. Phys [] de la Torre A C and Iguain J L 998 Am. J. Phys [3] Ohanian H C 986 Am. J. Phys

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