The Dirac formalism

of quantum mechanics

Anders BlomDivision of Solid State Theory

Department of Physics, Lund UniversitySolvegatan 14 A, S–223 62 Lund, Sweden

Anders.Blom@teorfys.lu.se

October 1, 2002

1 Introduction

The introductory courses in quantum mechanics usually work within the wave formalism, andexpress everything in real-space wavefunctions. This is probably partly due to a desire to makeanalogs with classical physics, in particular optics, and partly due to the fact that such coursesoften follow the historical development of the field.

If one instead introduces quantum mechanics from a more fundamental perspective, by formu-lating a few basic postulates, one finds, among many other things, that much of the formalismbecomes 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 workingwith three-dimensional wavefunctions, integrals and differential operators, the formalism insteadclosely resembles that of Euclidian vector spaces, and we can utilize well-known tools from linearalgebra.

In this short document we shall review the basic properties of states and operators in theformalism developed by the English physicist P. A. M. Dirac, which is the most convenient repre-sentation of this latter approach. I fully admit that the material is essentially just copied from theexcellent 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 notdue to my laziness, but instead indicates a very fundamental principle. The physics of quantummechanics – and of course also other fields of physics and the nature of the universe itself – isindependent of how we choose to formulate it mathematically. What we study here is merely onepossible way to represent the laws of physics, in a way which is consistent with all experimentswhich we perform to draw conclusions about the state of the physical reality surrounding us.

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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 ora ket α〉 belonging to a complex vector (ket) space. The ket is postulated to contain completeinformation of the physical state, i.e. it holds the answer to all conceivable measurements wecan perform on the system. Another way to characterize the state is therefore to say that it is acollection of all quantum labels (or quantum numbers), collectively represented by α in the ketabove, which describe the physical state we wish to consider.

Kets can be added, such that the sum α〉+ β〉 is just another ket γ〉, and so is also c α〉 =α〉 c, where c is a complex number. In fact, α〉 and c α〉 are postulated to represent to same

physical state, unless c = 0 in which case c α〉 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 anoperator A acts on a ket α〉, the results is another ket. In some cases

A α〉 = a α〉 (1)

where a is just a complex number. In this case, the ket α〉 is said to be an eigenket of theoperator A and a is the eigenvalue. Often the eigenkets are labeled by the eigenvalue itself, a〉.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 spanthe entire vector space. In other words, any arbitrary ket β〉 can be expanded in the eigenketsa〉 of an operator A as

β〉 =∑

a

ca a〉 , (2)

where ca are complex coefficients (numbers). The eigenkets are complete and orthogonal (seeproof 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 expansionis unique if the basis vectors are all linearly independent. Keep in mind, however, that unlikethe Euclidian vector spaces often considered in the basic mathematics courses, our vector space iscomplex.

We now postulate that to each ket α〉 in the ket space, there exists a corresponding quantitycalled a bra, denoted 〈α ,

α〉 ↔ 〈α .

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 α〉 is postulated to be c∗ 〈α :

c∗ 〈α ↔ c α〉 .

One can form an inner product between two kets α〉 and β〉, which we write as(〈α

)·(β〉)

= 〈α β〉 .

2

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

〈α β〉 = 〈β α〉∗ . (3)

From this follows that 〈α α〉 is a real number, and we postulate that it is positive, 〈α α〉 ≥ 0,where the equality sign only holds for the null ket. As a natural extension of the usual scalarproduct, two kets α〉 and β〉 are said to be orthogonal if

〈α β〉 = 0.

Finally, we shall for the present discussion assume that the kets always can be normalized bydividing by its norm

√〈α α〉, i.e. we require that 〈α α〉 < ∞. 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 deeperdiscussion of operators. Some properties of operators are rather trivial (although important, froma 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 underaddition. Such details can be found in the book by Sakurai. Furthermore, we shall in this paperonly consider linear operators, such that X

(a α〉+b β〉

)= aX α〉+bX β〉 for complex numbers

a and b.More interestingly, and important to remember, is that operators are (in general, there are

exceptions) non-commutative under multiplication, i.e.

XY 6= Y X (4)

They are, however, associative: X(Y Z) = (XY )ZWe 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 A† which acts in the mirror image bra space from the right:

A α〉 ↔ 〈α A†. (5)

The operator A† is named the Hermitian conjugate of A. An operator is called Hermitian orself-adjoint if A† = A.

Later on we will show how to represent operators as matrices, and that the Hermitian conjugatedoperator corresponds to the transposed complex conjugated matrix. It is therefore natural that

(XY )† = Y †X†, (6)

since this rule applies to the transpose of a matrix product. This property can easily be proved byusing the property (5).

There is another type of product, called outer product, between two kets α〉 and β〉, whichis defined as (

α〉)·(〈β

)= α〉 〈β .

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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 multipli-

cation, 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 α〉 β〉, and simply do notexist.

When an outer product acts on a ket, we can from the associative axiom change the order ofoperations, (

α〉 〈β)γ〉 = α〉

(〈β γ〉

). (7)

Now the quantity in parenthesis on the right hand side is an inner product, and hence a complexnumber δ = 〈β γ〉. This is why the outer product is an operator: when acting on a ket, it givesanother ket, in the case above this ket is δ α〉. An important property, which is easy to prove, isthat if

X = α〉 〈β then X† = β〉 〈α . (8)

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

〈α)·(A β〉

)=(〈α A

)·(β〉), (9)

and hence we can skip the parentheses and simply write

〈α | A | β〉.

Since 〈β A† is the bra corresponding to the ket A β〉, we have

〈α|A|β〉 = 〈α ·(A β〉

)={(〈β A†

)· α〉

}∗= 〈β|A†|α〉∗, (10)

where Eq. (5) was used. If A is Hermitian, then clearly 〈α|A|β〉 = 〈β|A|α〉∗.

2.3 Basis kets

In the previous section we postulated that the eigenkets of an operator span the entire vectorspace, and it was stated that the eigenkets of an operator form an orthogonal set. In fact, the setcan be taken to be orthonormal, since we can always normalize the kets. Hence,

〈a a′〉 = δaa′ (11)

for two eigenstates a〉 and a′〉 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 physicalobservables by Hermitian operators.

We postulated earlier that any state ket kan be expanded in the eigenstates of an operator,

β〉 =∑

a

ca a〉 . (12)

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Multiplying from the left with the bra 〈a′ we can determine the expansion coefficients, usingEq. (11), through the relation

〈a′ β〉 =∑

a

ca 〈a′ a〉 =∑

a

caδaa′ = ca′ . (13)

This means that we can write

β〉 =∑

a

a〉 〈a β〉 , (14)

which we immediately can compare with the corresponding expression for a vector V in Euclidianspace,

V =∑

i

ei

(ei ·V

),

for an orthonormal (note, this is necessary!) set of basis vectors {ei}.By the associative axiom one can alternatively consider a〉 〈a β〉 as a product of a number

〈a β〉 and a ket a〉, or the operator a〉 〈a acting on the ket β〉. Since β〉 is an arbitrary ket,we must have ∑

a

a〉 〈a = 1. (15)

This very important and useful property is called the completeness relation or closure, and isan immediate consequence of the fact that the eigenkets span the entire ket space. Since 1 is justthe identity operator, we can insert the left hand side of Eq. (15) where ever we wish, in order tosimplify or derive new expressions. For instance, if β〉 is normalized, we can show that

1 = 〈β β〉 = 〈β ·

(∑a

a〉 〈a

)β〉 =

∑a

|〈a β〉|2 =∑

a

|ca|2. (16)

Hence we can interpret |ca|2 = |〈a β〉|2 as the probability to find the state β〉 in the state a〉.We stated that outer products are operators. Let us therefore check what happens when we

apply the operator a′〉 〈a′ to the state β〉:(a′〉 〈a′

)· β〉 = a′〉 〈a′ a〉 〈a β〉 = ca′ a′〉 . (17)

Hence the operator a′〉 〈a′ selects only the part of the ket β〉 which is ”parallel” to a′〉, and itis therefore know as the projection operator, and labeled

Λa ≡ a〉 〈a , 1 =∑

a

Λa. (18)

The corresponding projection of a vector V on a direction defined by a vector e of unit magnitudein Euclidian vector space is well-known to be e(e ·V), and the ”projection operator” can in thiscase 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 thissection 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 infact derived parallel to Schrodinger’s wave-mechanical formalism. The two approaches could laterin a series of papers by Schrodinger, Bohr, Born and Heisenberg be shown to be equivalent1. Boththese formalisms are united in the abstract formalism due to Dirac we are considering here.

By using the closure relation (15) twice, we can write an operator X as

X =∑

a

∑a′

a〉 〈a|X|a′〉 〈a′ (19)

If there are N eigenkets a〉 of the operator A, there are N2 terms in the sum, and hence N2

numbers 〈a|A|a′〉. We can place these in a N ×N matrix, where the matrix elements are arrangedby columns and rows according to ⟨

a X a′⟩

↑ ↑row column

The matrix formed in this way is said to be a representation of the operator X. If we label theeigenstates of A as ai〉 with eigenvalues ai and i = 1..N , then we have explicitly

X.= X =

〈a1|X|a1〉 〈a1|X|a2〉 . . .

〈a2|X|a1〉 〈a2|X|a2〉 . . .

......

. . .

(20)

We will here and onwards use the symbol .= in the meaning ”represents”, and denote by X thematrix which represents an operator X.

Note that the matrix elements are just complex numbers. Although it should be obviousfrom the above presentation, it is certainly worth pointing out that the matrix representationdepends on which operator A we use as a basis.

If the matrix formalism is to be useful, we must retain all the well-known properties of matrixmanipulations. Remembering the rule (10), we can see that Hermitian conjugation simplycorresponds to transposing and complex conjugating the matrix. Since matrices in general donot 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.

Furthermore, the operator relation Z = XY becomes

〈a′|Z|a〉 = 〈a′|XY |a〉 =∑a′′

〈a′|X|a′′〉〈a′′|Y |a〉 (21)

1 See A. Messiah, Quantum Mehcanics, Dover (1999), pp. 45–49 for a short but illuminating discussion on theearly 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)between X and Y , and recovered the rule for matrix multiplication.

What happens if we now wish to act with the operator on a ket β〉 to form a new ket γ〉 =X β〉? The ket γ〉 can also be expanded in the eigenkets of A, and the expansion coefficients aregiven by

〈a γ〉 = 〈a|X|β〉 =∑a′

〈a|X|a′〉 〈a′ β〉 . (22)

But this is nothing but the usual rule for a matrix multiplying a column vector of numbers 〈a′ β〉.Hence we have found that while operators are represented by matrices, kets are represented bycolumn vectors, where each element is an expansion coefficient:

β〉 .=

〈a1 β〉

〈a2 β〉...

(23)

In a similar way it is not difficult to show that bras are represented by row vectors, with theminor change that all coefficients must be complex conjugated:

〈β .=(〈β a1〉 , 〈β a2〉 , . . .

)=(〈a1 β〉∗ , 〈a2 β〉∗ , . . .

). (24)

From this we immediately find that the inner product indeed corresponds closely to the usualscalar product,

〈β γ〉 .=(〈a1 β〉∗ , 〈a2 β〉∗ , . . .

)〈a1 γ〉

〈a2 γ〉...

=∑

i

〈β ai〉 〈ai γ〉 , (25)

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 A is used as a basis. Naturally, also theoperator A itself can be represented as a matrix, and in its own basis it is obvious that it becomesa diagonal matrix,

〈ai|A|aj〉 = aiδij . (26)

We can also write

A =∑

i

ai ai〉 〈ai =∑

i

aiΛai, (27)

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

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Example

The presentation above was rather heavy with a large number of indices. The principle is howeversimple, and with a short illustrative example, all should become clear. We shall use a spin 1

2 system,where the dimensionality of the ket space is 2; the only possible values of the spin projection areup and down. We shall use the eigenstates ↑〉 and ↓〉 of the Sz operator as basis states; theeigenvalues are ±1/2 in units where ~ = 1.

The eigenstates are clearly represented by the column matrices

↑〉 .=(

10

), ↓〉 .=

(01

),

and a general state γ〉 = a ↑〉+ b ↓〉 by

γ〉 .=(ab

).

Two operators which are very useful when discussing angular momentum are the lowering andraising (or ladder) operators

S+ ≡ ↑〉 〈↓ , S− ≡ ↓〉 〈↑ .

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

S+ ↑〉 = ↑〉 〈↓ ↑〉 = 0,

S+ ↓〉 = ↑〉 〈↓ ↓〉 = ↑〉 ,

S− ↑〉 = ↓〉 〈↑ ↑〉 = ↓〉 ,

S− ↓〉 = ↓〉 〈↑ ↓〉 = 0,

by the orthogonality of the basis kets. Thus S+ flips a down-spin into an up-state, and oppositefor S−, which takes an up-spin into a down-state. To consider the action of the operators on thegeneral spin state γ〉 is a useful exercise!

Finally, let us write down the matrix representation of the two ladder operators in the Sz basis.The only non-zero matrix elements are

〈↑ |S+| ↓〉 = 〈↑ ↑〉 〈↓ ↓〉 = 1, 〈↓ |S−| ↑〉 = 〈↓ ↓〉 〈↑ ↑〉 = 1,

and hence

S+.=(

0 10 0

)S−

.=(

0 01 0

)and we can check the previous result

S+ ↓〉 .=(

0 10 0

)(01

)=(

10

).= ↑〉 .

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4 Measurements

Although the considerations of quantum mechanical measurements are not particular to the Diracformulation, there appear a few postulates which are worth mentioning in the same context as therest of the formalism. We shall however be rather brief; a thorough discussion on the measure-ment process can be found in Sakurai, Chapter 1.4, which also includes details on the importantuncertainty relation. Another concept closely related to measurements is that of compatibleor 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 whichno information is given. Before any measurement is carried out, the only thing we can say about itsstate β〉 is that is can be expressed as a linear combination of the eigenstates of some observableA, represented by the operator A with eigenvalues a:

β〉 =∑

a

ca a〉 . (28)

We now make two postulates regarding the measurement process. First, we claim that theonly possible outcomes of a measurement of the observable A are the eigenvalues a, and that theprobability of obtaining the result a is |ca|2 = | 〈β a〉 |2. Since

∑a |ca|2 = 1 as was shown above,

the probabilities add up to unity, which is reassuring. This so-called probabilistic interpretationlies 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 jumpsinto the eigenstate a〉. In result, any subsequent measurements will yield the eigenvalue a withcertainty.

One must however be a bit careful with the probabilistic interpretation. A single measurementon a quantum mechanical state yields only one result, and does not give us any possibility todetermine the coefficients ca. For this, one must instead perform a series of measurements on anensemble of identically prepared systems. In this case, one may register with what probabil-ity each eigenvalue a appears, and hence determine |ca|2. Note, however, that it is impossible todetermine the phase of ca (remember that the coefficients are complex numbers), only its magni-tude. Furthermore, if a long series of measurements are performed, we can define the expectationvalue

〈A〉 ≡ 〈β|A|β〉 =∑

a

a|ca|2. (29)

Note that the expectation value depends on the particular state β〉. The definition of the expec-tation 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 itis expressed in. It is therefore of interest to try to relate two matrices expressed in different basesto eachother. In Euclidian vector space, where one may consider any orthonormal set of vectors asa 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

UU† = U†U = 1 (30)

Hence U−1 = U†.Assume now that we have two non-commuting operators2 A and B. One can then prove, that

the basis kets b〉 of B can always be expressed in the basis kets a〉 of A through some unitaryoperator U (for a proof, see Sakurai p. 37):

bi〉 = U ai〉 . (31)

To find the matrix representation U of U we can merely use Eq. (31) to rewrite the matrix elementin the A-representation:

UAij = 〈ai|U |aj〉 = 〈aj bi〉 . (32)

The next step is to express an arbitrary state ket γ〉, initially given in the ”old” basis of A as

γ〉 =∑

i

cai ai〉 , (33)

in the ”new” basis of B. Inserting the unit operator (15) once more, we get

〈bj γ〉 =∑

i

〈bj ai〉 〈ai γ〉 =∑

i

〈aj |U†|ai〉 〈ai γ〉 (34)

where we used the bra version of Eq. (31), 〈bj = 〈aj U†. The expression above is simply the matrixU† multiplying the column vector of the expansion coefficients ca. Hence the vector representationof the state ket in the new basis is found by multiplying the vector representation in the old basisby 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

(which are transformed into the new basis by U), is completely analogous to the fact that inEuclidian vector space, the coordinates of vectors rotate clock-wise if we rotate the basis vectorscounter-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 matricesrepresenting a general operator X in the new basis. By inserting two closure relations and usingEq. (31) twice, we get the matrix element in the new basis

〈bi|X|bj〉 =∑

n

∑m

〈bi an〉 〈an|X|am〉 〈am bj〉

=∑

n

∑m

〈ai|U†|an〉〈an|X|am〉〈am|U |aj〉(35)

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

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which can be identified as the matrix product3

XB = U†XAU. (36)

In linear algebra this is known as a similarity transformation.A useful exercise is to prove that the trace

TrX =∑

i

Xii =∑

i

〈ai|X|ai〉 (37)

of a matrix X corresponding to an operator X, is in fact independent of which basis is used torepresent it. As a consequence, since we can always represent an operator in its own eigenstates,

TrX =∏

i

xi (38)

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 Sx.From the Stern–Gerlach experiments (or from the general theory of angular momentum), it is

known that the eigenvalues of the Sx operator are also ±1/2 (with ~ = 1), and the eigenkets are

↑〉x =1√2

( ↑〉z + ↓〉z) ,

↓〉x =1√2

( ↑〉z − ↓〉z) ,(39)

where ↑〉z and ↓〉z are the eigenkets of Sz. Hence the unitary transformation is given by(↑x

↓x

)=

1√2

(1 11 −1

)(↑z

↓z

).

It is a useful exercise to check that the so-defined matrix U is indeed unitary. An alternative wayto determine the transformation matrix would be to diagonalize the matrix representation of theoperator Sx in the Sz basis,

Sx.=

12

(0 11 0

)z

which is also left as an exercise. Given the eigenvectors (which of course are nothing but thosepresented in Eq. (39)), the transformation matrix is found by placing the eigenvectors as thecolumns of U .

3 It is essential to keep in mind that this relationships depends on that bi〉 = U ai〉; if the relationship was

the opposite (sometimes it is easier to express the old basis in the new one), i.e. ai〉 = V bi〉, the matrixtransformation becomes XB = V XAV †.

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Let us now transform the operators S± to the new basis. U† is trivially obtained from U , andwe arrive at the matrix representation

U†S+U =12

(1 11 −1

)(0 10 0

)z

(1 11 −1

)=

12

(1 −11 −1

)x

,

U†S−U =12

(1 11 −1

)(0 01 0

)z

(1 11 −1

)=

12

(1 1−1 −1

)x

.

As a check, we can now evaluate

12

(Sx

+ + Sx−

)=

12

(1 00 −1

)x

,

which indeed is expected, since we know that S± = Sx ± iSy, and thus 12 (S+ + S−) = Sx 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 makethe connection to wave formalism. For simplicity we consider only the one-dimensional case, butthe results are easily generalized to three dimensions.

Although we have never really used it, it has been assumed until now that all states arenormalizable. An important group of states which fall outside this category are the eigenstates x〉of the position operator x. But even if the norm is infinite, we can always define the eigenvalue xfrom x x〉 = x x〉, 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

a〉 〈a →∫ ∞

−∞x〉 〈x dx. (40)

Hence the expansion of an arbitrary state is given by

α〉 =∫ ∞

−∞x〉 〈x α〉 dx. (41)

Let us compare this expression with an expansion such as Eq. (14), where we found that| 〈a β〉 |2 was the probability to find the state β〉 in the state a〉. If we make the analogousinterpretation now, we find that | 〈x α〉 |2dx is the probability to observe the particle within thesmall interval dx. Hence it is perfectly natural to define the wavefunction ψα(x) for the stateα〉 as the inner product of the state with the position eigenket at x,

ψα(x) ≡ 〈x α〉 , (42)

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

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From this follows immediately all the familiar rules of wave mechanics. As an example, considerthe inner product between two wavefunctions ψα(x) and ψβ(x) corresponding to two differencestates. By inserting the closure (40), we get

〈α β〉 =∫ ∞

−∞〈α x〉 〈x β〉 dx =

∫ ∞

−∞ψ∗α(x)ψβ(x) dx. (43)

Hence we see that the inner product measures the overlap between the two wavefunctions. Thisis in perfect accordance with our earlier stated general interpretation of 〈α β〉, that is measuresthe probability amplitude of state β〉 to be found in the state α〉.

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