The Dirac Formalism Of Quantum Mechanics

The Dirac formalism of quantum mechanics Anders Blom Division of Solid State Theory Department of Physics, Lund University S¨olvegatan 14 A, S–223 62 Lund, Sweden [email protected] October 1, 2002

1

Introduction

The introductory courses in quantum mechanics usually work within the wave formalism, and express everything in real-space wavefunctions. This is probably partly due to a desire to make analogs with classical physics, in particular optics, and partly due to the fact that such courses often follow the historical development of the field. If one instead introduces quantum mechanics from a more fundamental perspective, by formulating a few basic postulates, one finds, among many other things, that much of the formalism becomes independent of any particular representation (such as the three-dimensional real space), and that spin – which has no classical analogue – can be introduced effortlessly. Instead of working with three-dimensional wavefunctions, integrals and differential operators, the formalism instead closely resembles that of Euclidian vector spaces, and we can utilize well-known tools from linear algebra. In this short document we shall review the basic properties of states and operators in the formalism developed by the English physicist P. A. M. Dirac, which is the most convenient representation of this latter approach. I fully admit that the material is essentially just copied from the excellent book Modern Quantum Mechanics by J. J. Sakurai, published by Addison Wesley (1995), which follows exactly the path outlined above. I have skipped the proofs and some elaborations, and tried to make some parts simpler, or more extensive, as I thought it was warranted. You will note that the term ”represented by” appears a lot in the text which follows. This is not due to my laziness, but instead indicates a very fundamental principle. The physics of quantum mechanics – and of course also other fields of physics and the nature of the universe itself – is independent of how we choose to formulate it mathematically. What we study here is merely one possible way to represent the laws of physics, in a way which is consistent with all experiments which we perform to draw conclusions about the state of the physical reality surrounding us.

1

2

The ket and bra spaces

2.1

States

In the Dirac formalism of quantum mechanics, a physical state is represented by a state vector or a ket αi belonging to a complex vector (ket) space. The ket is postulated to contain complete information of the physical state, i.e. it holds the answer to all conceivable measurements we can perform on the system. Another way to characterize the state is therefore to say that it is a collection of all quantum labels (or quantum numbers), collectively represented by α in the ket above, which describe the physical state we wish to consider. Kets can be added, such that the sum αi + βi is just another ket γi, and so is also c αi = αi c, where c is a complex number. In fact, αi and c αi are postulated to represent to same physical state, unless c = 0 in which case c αi is the null ket. An observable is defined as something we can measure, such as the angular momentum or the position of a particle. In our formalism, observables will be represented by operators. When an operator Aˆ acts on a ket αi, the results is another ket. In some cases Aˆ αi = a αi

(1)

where a is just a complex number. In this case, the ket αi is said to be an eigenket of the operator Aˆ and a is the eigenvalue. Often the eigenkets are labeled by the eigenvalue itself, ai. It is natural to use the name eigenstate for the physical state corresponding to an eigenket. A fundamental postulate of great importance is that the eigenkets of an observable span the entire vector space. In other words, any arbitrary ket βi can be expanded in the eigenkets ai of an operator Aˆ as X βi = ca ai , (2) a

where ca are complex coefficients (numbers). The eigenkets are complete and orthogonal (see proof in Sakurai, pp. 17–18) and hence the expansion is unique. This is the first place where we can see a strong and clear parallel to the world of linear algebra, where we know that any vector can be expressed in a set of basis vectors, and that the expansion is unique if the basis vectors are all linearly independent. Keep in mind, however, that unlike the Euclidian vector spaces often considered in the basic mathematics courses, our vector space is complex. We now postulate that to each ket αi in the ket space, there exists a corresponding quantity called a bra, denoted hα , αi ↔ hα . All the bras together span a bra space, which can be viewed as a mirror image of the ket space. It is important to remember that the bra corresponding to c αi is postulated to be c∗ hα : c∗ hα ↔ c αi . One can form an inner product between two kets αi and βi, which we write as

hα · βi = hα βi .

2

The inner product is a complex number, and is essentially equivalent to the scalar product of two vectors in a usual Euclidian vector space. There is however one important difference. For two usual vectors a and b, the scalar products a · b = b · a, but since the inner product in general is complex, we postulate the following property: ∗

hα βi = hβ αi .

(3)

From this follows that hα αi is a real number, and we postulate that it is positive, hα αi ≥ 0, where the equality sign only holds for the null ket. As a natural extension of the usual scalar product, two kets αi and βi are said to be orthogonal if hα βi = 0. Finally, we shall forpthe present discussion assume that the kets always can be normalized by dividing by its norm hα αi, i.e. we require that hα αi < ∞. We will not consider continuous spectra (which are not normalizable), except for a few specific cases later on.

2.2

Operators

Having revised the basic properties of kets, bras and states, we now turn our attention to a deeper discussion of operators. Some properties of operators are rather trivial (although important, from a fundamental point of view), such as how to determine if two operators are equal, the null operator (which produces a null ket) and the fact that operators are commutative and associative under addition. Such details can be found in the book by Sakurai. Furthermore, we shall in this paper
ˆ a αi+b βi = aX ˆ αi+bX ˆ βi for complex numbers only consider linear operators, such that X a and b. More interestingly, and important to remember, is that operators are (in general, there are exceptions) non-commutative under multiplication, i.e. ˆ Yˆ 6= Yˆ X ˆ X

(4)

ˆ Yˆ Z) ˆ = (X ˆ Yˆ )Zˆ They are, however, associative: X( We saw earlier that an operator acts on a ket from the left, but how to operate on a bra? For this, we introduce an operator denoted Aˆ† which acts in the mirror image bra space from the right: Aˆ αi ↔ hα Aˆ† .

(5)

ˆ An operator is called Hermitian or The operator Aˆ† is named the Hermitian conjugate of A. † ˆ ˆ self-adjoint if A = A. Later on we will show how to represent operators as matrices, and that the Hermitian conjugated operator corresponds to the transposed complex conjugated matrix. It is therefore natural that ˆ Yˆ )† = Yˆ † X ˆ †, (X

(6)

since this rule applies to the transpose of a matrix product. This property can easily be proved by using the property (5). There is another type of product, called outer product, between two kets αi and βi, which is defined as

αi · hβ = αi hβ . 3

Unlike the inner product, the outer product is not a complex number but an operator. The final postulate regarding kets and bras is the so-called associative axiom of multiplication, which states that as long as we do not make any ”illegal” combinations of bras and kets, the associate property holds. Illegal combinations are for instance αi βi, and simply do not exist. When an outer product acts on a ket, we can from the associative axiom change the order of operations,

αi hβ γi = αi hβ γi . (7) Now the quantity in parenthesis on the right hand side is an inner product, and hence a complex number δ = hβ γi. This is why the outer product is an operator: when acting on a ket, it gives another ket, in the case above this ket is δ αi. An important property, which is easy to prove, is that if ˆ = αi hβ X

ˆ † = βi hα . then X

(8)

A second, perhaps even more important, illustration of the associative axiom can be seen by using a different order of kets and including an operator. By the associative axiom,



hα · Aˆ βi = hα Aˆ · βi , (9) and hence we can skip the parentheses and simply write hα | Aˆ | βi. Since hβ Aˆ† is the bra corresponding to the ket Aˆ βi, we have o∗
n
ˆ hα|A|βi = hα · Aˆ βi = hβ Aˆ† · αi = hβ|Aˆ† |αi∗ ,

(10)

ˆ ˆ ∗. where Eq. (5) was used. If Aˆ is Hermitian, then clearly hα|A|βi = hβ|A|αi

2.3

Basis kets

In the previous section we postulated that the eigenkets of an operator span the entire vector space, and it was stated that the eigenkets of an operator form an orthogonal set. In fact, the set can be taken to be orthonormal, since we can always normalize the kets. Hence, ha a0 i = δaa0

(11)

ˆ for two eigenstates ai and a0 i of an operator A. For the particular case of Hermitian operators, it is further possible to show that all eigenvalues are real. Since physical properties are always required to be real, it is natural to represent physical observables by Hermitian operators. We postulated earlier that any state ket kan be expanded in the eigenstates of an operator, X βi = ca ai . (12) a

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Multiplying from the left with the bra ha0 we can determine the expansion coefficients, using Eq. (11), through the relation X X ha0 βi = ca ha0 ai = ca δaa0 = ca0 . (13) a

a

This means that we can write βi =

X

ai ha βi ,

(14)

a

which we immediately can compare with the corresponding expression for a vector V in Euclidian space, X
ˆi e ˆi · V , e V= i

for an orthonormal (note, this is necessary!) set of basis vectors {ˆ ei }. By the associative axiom one can alternatively consider ai ha βi as a product of a number ha βi and a ket ai, or the operator ai ha acting on the ket βi. Since βi is an arbitrary ket, we must have X ai ha = 1. (15) a

This very important and useful property is called the completeness relation or closure, and is an immediate consequence of the fact that the eigenkets span the entire ket space. Since 1 is just the identity operator, we can insert the left hand side of Eq. (15) where ever we wish, in order to simplify or derive new expressions. For instance, if βi is normalized, we can show that ! X X X 2 1 = hβ βi = hβ · ai ha βi = |ha βi| = |ca |2 . (16) a

a

a

2

Hence we can interpret |ca |2 = |ha βi| as the probability to find the state βi in the state ai. We stated that outer products are operators. Let us therefore check what happens when we apply the operator a0 i ha0 to the state βi:
a0 i ha0 · βi = a0 i ha0 ai ha βi = ca0 a0 i . (17) Hence the operator a0 i ha0 selects only the part of the ket βi which is ”parallel” to a0 i, and it is therefore know as the projection operator, and labeled X Λa ≡ ai ha , 1= Λa . (18) a

ˆ of unit magnitude The corresponding projection of a vector V on a direction defined by a vector e ˆ(ˆ in Euclidian vector space is well-known to be e e · V), and the ”projection operator” can in this ˆ(ˆ case be thought of as e e·).

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3

The matrix formalism

We have already several times, without reflecting over why, called our states ”vectors”. In this section we shall present even stronger analogies between the ket space formalism and linear algebra, and represent states as (complex) vectors and operators as matrices. The matrix formulation of quantum mechanics, due to Heisenberg, Jordan and Born, was in fact derived parallel to Schr¨ odinger’s wave-mechanical formalism. The two approaches could later in a series of papers by Schr¨ odinger, Bohr, Born and Heisenberg be shown to be equivalent1 . Both these formalisms are united in the abstract formalism due to Dirac we are considering here. ˆ as By using the closure relation (15) twice, we can write an operator X XX ˆ 0 i ha0 ˆ= ai ha|X|a (19) X a

a0

ˆ there are N 2 terms in the sum, and hence N 2 If there are N eigenkets ai of the operator A, ˆ 0 i. We can place these in a N × N matrix, where the matrix elements are arranged numbers ha|A|a by columns and rows according to

 ˆ a0 a X ↑ ↑ row column ˆ If we label the The matrix formed in this way is said to be a representation of the operator X. ˆ eigenstates of A as ai i with eigenvalues ai and i = 1..N , then we have explicitly   ˆ 1 i ha1 |X|a ˆ 2i . . . ha1 |X|a   .  ˆ 1 i ha2 |X|a ˆ 2 i . . . ˆ= X X = ha2 |X|a (20)    .. .. .. . . . . We will here and onwards use the symbol = in the meaning ”represents”, and denote by X the ˆ matrix which represents an operator X. Note that the matrix elements are just complex numbers. Although it should be obvious from the above presentation, it is certainly worth pointing out that the matrix representation depends on which operator Aˆ we use as a basis. If the matrix formalism is to be useful, we must retain all the well-known properties of matrix manipulations. Remembering the rule (10), we can see that Hermitian conjugation simply corresponds to transposing and complex conjugating the matrix. Since matrices in general do not commute under multiplication, the same applies to operators, as was stated already earlier. We also find that the rule (6) holds, by the arguments presented just before this equation. ˆ Yˆ becomes Furthermore, the operator relation Zˆ = X X ˆ ˆ Yˆ |ai = ˆ 00 iha00 |Yˆ |ai ha0 |Z|ai = ha0 |X ha0 |X|a (21) a00

1

See A. Messiah, Quantum Mehcanics, Dover (1999), pp. 45–49 for a short but illuminating discussion on the early development of quantum mechanics during the years 1923–30.

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for each element in the matrix Z. We just inserted the identity operator in the form of Eq. (15) ˆ and Yˆ , and recovered the rule for matrix multiplication. between X What happens if we now wish to act with the operator on a ket βi to form a new ket γi = ˆ and the expansion coefficients are ˆ βi? The ket γi can also be expanded in the eigenkets of A, X given by X ˆ ˆ 0 i ha0 βi . ha γi = ha|X|βi = ha|X|a (22) a0

But this is nothing but the usual rule for a matrix multiplying a column vector of numbers ha0 βi. Hence we have found that while operators are represented by matrices, kets are represented by column vectors, where each element is an expansion coefficient:   ha1 βi   .   βi = ha2 βi (23)   .. . In a similar way it is not difficult to show that bras are represented by row vectors, with the minor change that all coefficients must be complex conjugated:     . ∗ ∗ hβ = hβ a1 i , hβ a2 i , . . . = ha1 βi , ha2 βi , . . . . (24) From this we immediately find that the inner product indeed corresponds closely to the usual scalar product,   ha1 γi  X   .   ∗ ∗ hβ γi = ha1 βi , ha2 βi , . . . ha2 γi = (25) hβ ai i hai γi ,   i .. . where we again recognize the identity operator in the last term. As we pointed out earlier, the matrix representation of the operators (and of course also the kets and bras) depends intimately on which operator Aˆ is used as a basis. Naturally, also the operator Aˆ itself can be represented as a matrix, and in its own basis it is obvious that it becomes a diagonal matrix, ˆ j i = ai δij . hai |A|a

(26)

We can also write Aˆ =

X

ai ai i hai =

i

X i

where Λai is the projection operator defined in Eq. (18).

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ai Λai ,

(27)

Example The presentation above was rather heavy with a large number of indices. The principle is however simple, and with a short illustrative example, all should become clear. We shall use a spin 12 system, where the dimensionality of the ket space is 2; the only possible values of the spin projection are up and down. We shall use the eigenstates ↑i and ↓i of the Sˆz operator as basis states; the eigenvalues are ±1/2 in units where ~ = 1. The eigenstates are clearly represented by the column matrices     . 1 . 0 ↑i = , ↓i = , 0 1 and a general state γi = a ↑i + b ↓i by . γi =

  a . b

Two operators which are very useful when discussing angular momentum are the lowering and raising (or ladder) operators Sˆ+ ≡ ↑i h↓ ,

Sˆ− ≡ ↓i h↑ .

† Note that these operators are not Hermitian; in fact Sˆ+ = Sˆ− . Let us consider their action on the eigenkets:

Sˆ+ ↑i = ↑i h↓ ↑i = 0, Sˆ+ ↓i = ↑i h↓ ↓i = ↑i , Sˆ− ↑i = ↓i h↑ ↑i = ↓i , Sˆ− ↓i = ↓i h↑ ↓i = 0, by the orthogonality of the basis kets. Thus Sˆ+ flips a down-spin into an up-state, and opposite for Sˆ− , which takes an up-spin into a down-state. To consider the action of the operators on the general spin state γi is a useful exercise! Finally, let us write down the matrix representation of the two ladder operators in the Sˆz basis. The only non-zero matrix elements are h↑ |Sˆ+ | ↓i = h↑ ↑i h↓ ↓i = 1,

h↓ |Sˆ− | ↑i = h↓ ↓i h↑ ↑i = 1,

and hence . Sˆ+ =





0 0

1 0

. Sˆ− =



0 1

 0 0

and we can check the previous result . Sˆ+ ↓i =



0 0

    1 0 1 . = = ↑i . 0 1 0

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4

Measurements

Although the considerations of quantum mechanical measurements are not particular to the Dirac formulation, there appear a few postulates which are worth mentioning in the same context as the rest of the formalism. We shall however be rather brief; a thorough discussion on the measurement process can be found in Sakurai, Chapter 1.4, which also includes details on the important uncertainty relation. Another concept closely related to measurements is that of compatible or commuting operators. We will however not overload our discussion by including this topic, but refer the interested reader to the literature. Assume that we are presented with a quantum mechanical system, such as a particle, of which no information is given. Before any measurement is carried out, the only thing we can say about its state βi is that is can be expressed as a linear combination of the eigenstates of some observable A, represented by the operator Aˆ with eigenvalues a: X βi = ca ai . (28) a

We now make two postulates regarding the measurement process. First, we claim that the only possible outcomes of a measurement of the observable A are P the eigenvalues a, and that the probability of obtaining the result a is |ca |2 = | hβ ai |2 . Since a |ca |2 = 1 as was shown above, the probabilities add up to unity, which is reassuring. This so-called probabilistic interpretation lies at the heart of quantum mechanics, but was the cause of much debate in its early days. The second postulate is, that immediately when the measurement is performed, the state jumps into the eigenstate ai. In result, any subsequent measurements will yield the eigenvalue a with certainty. One must however be a bit careful with the probabilistic interpretation. A single measurement on a quantum mechanical state yields only one result, and does not give us any possibility to determine the coefficients ca . For this, one must instead perform a series of measurements on an ensemble of identically prepared systems. In this case, one may register with what probability each eigenvalue a appears, and hence determine |ca |2 . Note, however, that it is impossible to determine the phase of ca (remember that the coefficients are complex numbers), only its magnitude. Furthermore, if a long series of measurements are performed, we can define the expectation value X ˆ hAi ≡ hβ|A|βi = a|ca |2 . (29) a

Note that the expectation value depends on the particular state βi. The definition of the expectation value clearly agrees with our usual understanding of the average measured value.

5

Change of basis

It was pointed out earlier that the matrix representation of an operator depends on which basis it is expressed in. It is therefore of interest to try to relate two matrices expressed in different bases to eachother. In Euclidian vector space, where one may consider any orthonormal set of vectors as a basis, this is done by orthogonal rotation matrices. In the complex ket space, one instead uses

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unitary matrices (which of course represent unitary operators), defined by the property U U † = U †U = 1

(30)

Hence U −1 = U † . ˆ One can then prove, that Assume now that we have two non-commuting operators2 Aˆ and B. ˆ can always be expressed in the basis kets ai of Aˆ through some unitary the basis kets bi of B ˆ (for a proof, see Sakurai p. 37): operator U ˆ ai i . bi i = U

(31)

ˆ we can merely use Eq. (31) to rewrite the matrix element To find the matrix representation U of U ˆ in the A-representation: A ˆ |aj i = haj bi i . Uij = hai |U

(32)

The next step is to express an arbitrary state ket γi, initially given in the ”old” basis of Aˆ as X γi = cai ai i , (33) i

ˆ Inserting the unit operator (15) once more, we get in the ”new” basis of B. X X ˆ † |ai i hai γi hbj γi = hbj ai i hai γi = haj |U i

(34)

i

ˆ † . The expression above is simply the matrix where we used the bra version of Eq. (31), hbj = haj U † U multiplying the column vector of the expansion coefficients ca . Hence the vector representation of the state ket in the new basis is found by multiplying the vector representation in the old basis by the matrix U † ; {New} = U † {Old}, as Sakurai puts it symbolically. The fact that the state kets are transformed in the opposite way, compared to the basis kets ˆ ), is completely analogous to the fact that in (which are transformed into the new basis by U Euclidian vector space, the coordinates of vectors rotate clock-wise if we rotate the basis vectors counter-clockwise, to keep the actual physical vector (i.e. the state) fixed in space. Once we know how to transform the state vectors, we also need to evaluate the matrices ˆ in the new basis. By inserting two closure relations and using representing a general operator X Eq. (31) twice, we get the matrix element in the new basis XX ˆ ji = ˆ m i ham bj i hbi |X|b hbi an i han |X|a n

=

n

2

m

XX

ˆ † |an ihan |X|a ˆ m iham |U ˆ |aj i hai |U

m

If they commute, they have a common set of eigenstates, and this situation is not interesting.

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(35)

which can be identified as the matrix product3 X B = U † X A U. In linear algebra this is known as a similarity transformation. A useful exercise is to prove that the trace X X ˆ ii Tr X = Xii = hai |X|a i

(36)

(37)

i

ˆ is in fact independent of which basis is used to of a matrix X corresponding to an operator X, represent it. As a consequence, since we can always represent an operator in its own eigenstates, Y Tr X = xi (38) i

ˆ where xi are the eigenvalues of the operator X.

Example As an example, consider the spin 12 system of the previous section again. Assume we wish to change our basis to the eigenstates of Sˆx . From the Stern–Gerlach experiments (or from the general theory of angular momentum), it is known that the eigenvalues of the Sˆx operator are also ±1/2 (with ~ = 1), and the eigenkets are 1 ↑ix = √ ( ↑iz + ↓iz ) , 2 1 ↓ix = √ ( ↑iz − ↓iz ) , 2

(39)

where ↑iz and ↓iz are the eigenkets of Sˆz . Hence the unitary transformation is given by      1 1 1 ↑x ↑z =√ . ↓x ↓z 2 1 −1 It is a useful exercise to check that the so-defined matrix U is indeed unitary. An alternative way to determine the transformation matrix would be to diagonalize the matrix representation of the operator Sˆx in the Sˆz basis,   . 1 0 1 Sˆx = 2 1 0 z which is also left as an exercise. Given the eigenvectors (which of course are nothing but those presented in Eq. (39)), the transformation matrix is found by placing the eigenvectors as the columns of U . 3

ˆ ai i; if the relationship was It is essential to keep in mind that this relationships depends on that bi i = U the opposite (sometimes it is easier to express the old basis in the new one), i.e. ai i = Vˆ bi i, the matrix transformation becomes X B = V X A V † .

11

Let us now transform the operators Sˆ± to we arrive at the matrix representation   1 1 1 0 U † Sˆ+ U = 1 −1 0 2   1 1 1 0 †ˆ U S− U = 1 2 1 −1

the new basis. U † is trivially obtained from U , and      1 1 −1 1 1 1 = , 0 z 1 −1 2 1 −1 x      1 1 0 1 1 1 . = 0 z 1 −1 2 −1 −1 x

As a check, we can now evaluate  1 1  ˆx x S+ + Sˆ− = 2 2



 1 0 , 0 −1 x

which indeed is expected, since we know that Sˆ± = Sˆx ± iSˆy , and thus 12 (Sˆ+ + Sˆ− ) = Sˆx which is diagonal in its own basis, with the eigenvalues appearing on the diagonal.

6

The real-space wavefunction

As a final point in this expose of the basic properties of kets and their relatives, it is time to make the connection to wave formalism. For simplicity we consider only the one-dimensional case, but the results are easily generalized to three dimensions. Although we have never really used it, it has been assumed until now that all states are normalizable. An important group of states which fall outside this category are the eigenstates xi of the position operator x ˆ. But even if the norm is infinite, we can always define the eigenvalue x from x ˆ xi = x xi, which simply is the position in real space. For the interested reader, Sakurai discusses the continuous spectrum in detail in Chapters 1.6– 1.7. The only result we really need from that is that the closure relation now becomes Z ∞ ai ha → xi hx dx. (40) −∞

Hence the expansion of an arbitrary state is given by Z ∞ αi = xi hx αi dx.

(41)

−∞

Let us compare this expression with an expansion such as Eq. (14), where we found that | ha βi |2 was the probability to find the state βi in the state ai. If we make the analogous interpretation now, we find that | hx αi |2 dx is the probability to observe the particle within the small interval dx. Hence it is perfectly natural to define the wavefunction ψα (x) for the state αi as the inner product of the state with the position eigenket at x, ψα (x) ≡ hx αi ,

(42)

since this gives exactly the same probability interpretation as above, which we are familiar with from wave mechanics. In this way, the wavefunction receives no special status, but is simply yet another expansion coefficient, viz. the projection of the state onto the position eigenstates. 12

From this follows immediately all the familiar rules of wave mechanics. As an example, consider the inner product between two wavefunctions ψα (x) and ψβ (x) corresponding to two difference states. By inserting the closure (40), we get Z ∞ Z ∞ hα xi hx βi dx = ψα∗ (x)ψβ (x) dx. (43) hα βi = −∞

−∞

Hence we see that the inner product measures the overlap between the two wavefunctions. This is in perfect accordance with our earlier stated general interpretation of hα βi, that is measures the probability amplitude of state βi to be found in the state αi.

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