aa r X i v : . [ phy s i c s . e d - ph ] A ug Do bras and kets have dimensions?
Claude Semay ∗ and Cintia Willemyns † Service de Physique Nucl´eaire et Subnucl´eaire, Universit´e de Mons,UMONS Research Institute for Complex Systems, Place du Parc 20, 7000 Mons, Belgium (Dated: August 10, 2020)The bra and ket notation introduced by Dirac and the dimensional analysis are two powerful toolsfor the physicist. Curiously, almost nothing is said about connections between these two topics inthe literature. We show here that bras and kets have dimensions. This could help students to graspa better comprehension of this abstract notation.
Keywords: Dirac notation, dimensional analysis
The bra and ket notation introduced by Dirac is widely used in many textbooks (see [1] for instance). Althoughnot devoid of problems and shortcomings [2], this formalism is very powerful to perform changes of basis or to handlespinor states, for instance. The usefulness of the dimensional analysis must no longer be demonstrated [3, 4], whetherit is to simply check formulas or to build new physical laws in difficult problems. It can be particularly suited inquantum mechanics which is not known for its very intuitive nature [5]. Nevertheless, very few information can befound about the dimensions of bras and kets in the literature. This is not a comfortable situation. Since bras and ketsare relevant physical quantities, they are expected to be characterized by units or dimensions [6]. The following textis a detailed analysis of a question found on the Internet: “Do bras and kets have dimensions?” [7]. It is shown thatthe question is relevant and that the answer is positive. This could help students to grasp a better comprehension ofthis abstract notation.In this paper, we will determine the dimensions of bras and kets for a one-body system in one dimension. But theresults can be easily extended to N -body systems in D dimensions. In the following, we will only use the dimensionsof mass, length and time: M , L and T . This is convenient for our purpose but there is some freedom in the choice ofthe basic dimensions [3, 8]. As usual, the dimension of a quantity Q is noted [ Q ], with the convention [ Q ] = 1 if Q isdimensionless. If x is the position and p the conjugate momentum, then[ x ] = L and [ p ] = M L T − , (1)with [ ~ ] = [ x ] [ p ] = M L T − . If | α i and | β i are kets of the Hilbert space considered, then[ h α | β i ] = [ h α | ] [ | β i ] and [ h α | β i ] = [ h β | α i ] . (2)The first relation stems from the fact the bracket h α | β i is the result of the action of the bra h α | on the ket | β i . Thesecond one is the result of the equality h α | β i = h β | α i ∗ since the complex conjugation does not change the dimensions.But there is a priori no reason to assume that [ | α i ] = [ h α | ]. We will come back to this point below. If the ket | ψ i represents a physical state, let us note [ | ψ i ] = K and [ h ψ | ] = B , where K and B are dimensional formulas. A physicalstate is always normalisable (but it could be built by a superposition of non-normalisable states like the wave packetof a free particle) [1]. With relations (2), the normalisation condition h ψ | ψ i = 1 implies that B = K − .The continuous position basis {| x i} is such that ˆ x | x i = x | x i , where ˆ x is the position operator. The wave functionin the position space Ψ( x, t ) is defined by [1] h x | Ψ i = h Ψ | x i ∗ = Ψ( x, t ) , (3)with [Ψ( x, t )] = L − / because of the normalisation condition Z + ∞−∞ | Ψ( x, t ) | dx = 1 . (4)Let us note [ h x | ] = X B and [ | x i ] = X K . Equation (3) implies X B K = X K K − = L − / . (5) ∗ E-mail: [email protected] ; ORCiD: 0000-0001-6841-9850 † E-mail: [email protected] ; ORCiD: 0000-0001-8114-0061
Typeset by REVTEXThe closure relation is given by [1] | ψ i = Z + ∞−∞ Ψ( y, t ) | y i dy with h x | y i = δ ( x − y ) , (6)where y is a position. The last equality with the Dirac delta function is called “Dirac orthonormality” in [1]. Equa-tions (6) imply respectively K = X K L / and X B X K = L − . (7)This does not bring any new information, since (7) can be deduced from (5). So, at this stage, relations exist between X B , X K and K , but their dimensional formulas are not fixed.Let us look at the continuous momentum basis {| p i} . It is such that ˆ p | p i = p | p i , where ˆ p is the momentum operatorwith [ˆ x, ˆ p ] = i ~ . The wave function in the momentum space Φ( p, t ) is defined by [1] h p | Ψ i = h Ψ | p i ∗ = Φ( p, t ) . (8)Φ( p, t ) is normalised because Ψ( x, t ) is normalised [1]. If we note [ h p | ] = P B and [ | p i ] = P K , the normalisationcondition implies P B K = P K K − = ( M L T − ) − / . (9)The corresponding closure relation, given by [1] | ψ i = Z + ∞−∞ Φ( q, t ) | q i dq with h p | q i = δ ( p − q ) , (10)where q is a momentum, does not bring supplementary information about dimensions, as in the case of the positionbasis.Using the Dirac notation, it is very easy to find the link between the position-space wave function Ψ( x, t ) and themomentum-space wave function Φ( p, t ): [1]Φ( p, t ) = Z + ∞−∞ h p | y i Ψ( y, t ) dy, (11)where h p | x i = h x | p i ∗ = 1 √ π ~ e − i p x/ ~ . (12)Simple manipulations show that (11) and (12) are compatible with (5) and (9), but with nothing new. The definitionof bases relying on other conjugate variables will lead to the same results.But a physical state | Ψ i can also be expanded in a discrete orthonormal basis {| n i} , [1] | Ψ i = ∞ X n =1 C n ( t ) | n i , (13)with C n ( t ) = h n | Ψ i and h n | m i = δ nm . (14)The normalisation condition gives h Ψ | Ψ i = ∞ X m,n =1 C m ( t ) ∗ C n ( t ) h m | n i = 1 . (15) It is our teaching experience that students are generally not aware that wave functions and Dirac delta function carry dimensions. So,it is useful to insist on this point.
This implies [ C m ( t ) ∗ ] [ C n ( t )] [ h m | n i ] = [ C n ( t )] [ h m | n i ] = 1, that is to say [ C n ( t )] = 1, thanks to the normalisationcondition for the discrete states. Noting [ | n i ] = N K and [ h n | ] = N B , (13) and its “bra” counterpart give immediately K = N K and B = N B = K − . (16)In fact, this is not very surprising since the basis states | n i are generally the eigenstates of a given operator ˆ A ,ˆ A | n i = a n | n i , with a definite physical content.So, the dimensions of the various bras and kets presented above cannot be univocally determined. The attributionof dimensions to a quantity is to some extent a matter of convention, provided the global coherence of the wholesystem is guaranteed [3, 8]. It is then not shocking to fix a convention for bras and kets. Two conditions appear quitenaturally.First, in order to simplify all the relations given above, it can be decided that [ | Ψ i ] = K = 1. As a consequence,a ket and its corresponding bra for a physical state are dimensionless. This is in complete agreement with the Diracnotation, which is by construction independent of any representation. But this is not the case for the continuous basisstates which refer explicitly to a given representation.Second, we can safely claim that there can be no difference between the physics with kets or with bras. For instance,the counterpart of (6) is h ψ | = Z + ∞−∞ Ψ( y, t ) ∗ h y | dy. (17)It seems then also quite natural to enforce that the dimensions for a ket and its corresponding bra are equal.Fortunately, the examination of formulas above shows that these two conventions are equivalent:[ h α | ] = [ | α i ] ⇔ K = 1 . (18)So, K = B = N K = N B = 1 , X B = X K = L − / and P B = P K = ( M L T − ) − / . (19)We conclude that the notion of dimensions is relevant for bras and kets. Our results are resumed in Table I. From theseproperties, [ h α | β i ] = [ h α | ] [ | β i ] = [ | α i ] [ h β | ] = [ h β | α i ]. This is compatible with the fact that the complex conjugationdoes not change the dimensions. TABLE I: Dimensions for bras and kets for a one-body system in one dimension. The two equivalent conventions are markedin bold. general property [ h α | β i ] = [ h α | ] [ | β i ] bra/ket [ h α | ] = [ | α i ] physical state [ | Ψ i ] = 1position basis [ | x i ] = L − / momentum basis [ | p i ] = ( M L T − ) − / discrete basis [ | n i ] = 1 All the results obtained above are easy to generalise for different situations. For a N -body system in a spacewith D dimensions, Ψ( r , . . . , r N ) = h r , . . . , r N | Ψ i and [Ψ( r , . . . , r N )] = L − ND/ . If a basis ket is associated witha dimensioned continuous variable, its dimension is determined by the corresponding Dirac orthonormality. For adiscrete basis, the corresponding kets are dimensionless.Finally, it could seem strange that the dimensions of a ket change with its status. But we can attempt a (bold)parallel with the expansion of a three-dimensional vector a = P i =1 a i e i in various bases. In a Cartesian basis,[ e x ] = [ e y ] = [ e z ] = 1, while in a spherical natural basis, [ e ρ ] = 1 and [ e θ ] = [ e φ ] = L . This is obvious by examiningthe corresponding metric in which the dimensions of the basis elements e i appear through the values of e i . Acknowledgments
One of the author (CS) would like to thank Thomas Brihaye for drawing his attention to the existence of reference [2],which was the starting motivation for this paper. This work was supported by the Fonds de la Recherche Scientifique- FNRS under Grant Number 4.4510.08. [1] Griffiths D J and Schroeter D F 2018
Introduction to Quantum Mechanics (Cambridge University Press, Cambridge)[2] Gieres F 2000 Mathematical surprises and Dirac’s formalism in quantum mechanics
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