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Dirac Notation 1 Vectors Physics 324, Fall 2001 Dirac Notation 1 Vectors 1.1 Inner product T Recall from linear algebra: we can represent a vector V as a column vector; then V y = (V )∗ is a row vector, and the inner product (another name for dot product) between two vectors is written as AyB = A∗B1 + A∗B2 + : (1) 1 2 ··· In conventional vector notation, the above is just A~∗ B~ . Note that the inner product of a vector with itself is positive definite; we can define the· norm of a vector to be V = pV V; (2) j j y which is a non-negative real number. (In conventional vector notation, this is V~ , which j j is the length of V~ ). 1.2 Basis vectors We can expand a vector in a set of basis vectors e^i , provided the set is complete, which means that the basis vectors span the whole vectorf space.g The basis is called orthonormal if they satisfy e^iye^j = δij (orthonormality); (3) and an orthonormal basis is complete if they satisfy e^ e^y = I (completeness); (4) X i i i where I is the unit matrix. (note that a column vectore ^i times a row vectore ^iy is a square matrix, following the usual definition of matrix multiplication). Assuming we have a complete orthonormal basis, we can write V = IV = e^ e^yV V e^ ;V (^eyV ) : (5) X i i X i i i i i ≡ i ≡ The Vi are complex numbers; we say that Vi are the components of V in the e^i basis. f g 1 1.3 Eigenvectors as basis vectors Sometimes it is convenient to choose as basis vectors the eigenvectors of a particular matrix. In quantum mechanics, measurable quantities correspond to hermitian operators; so here we will look at hermitian matrices. A hermitian matrix is one satisfying T M = My (M )∗ (hermitian) : (6) ≡ This just means that the components of a hermitian matrix satisfy Mij = Mji∗ . We say that the vector vn is an eigenvector of the matrix M if it satisfies Mvn = λnvn ; (7) where λn is a number called an eigenvalue of M. If M is hermitian, the eigenvalues λn are all real, and the eigenvectors may be taken to be orthonormal: vmy vn = δnm : (8) So we can take the vn to be our basis vectors, and write an arbitrary vector A in this basis as A = A v : (9) X n n n where the An are in general complex numbers. This is a convenient choice if we wish to know what is the action of the hermitian matrix M when it multiplies the vector A: MA = A Mv = A λ v : (10) X n n X n n n n n 2 Dirac notation for vectors Now let us introduce Dirac notation for vectors. We simply rewrite all the equations in the above section in terms of bras and kets. We replace V V ;V y V ;AyB A B : (11) ! j i ! h j ! h j i Suppose we have basis vector i , analogous to thee ^i, which form a complete orthonormal set: j i i j = δ (orthonormality) ij (12) hi j ii = 1 (completeness) ; Pi j ih j where 1 is the identity operator; it has the property 1 = for any . Then any vector V may be expanded in this basis asj i j i j i j i V = 1 V = i i V V i ;V i V : (13) X X i i j i j i i j ih j i ≡ i j i ≡ h j i 2 Note that V i = Vi∗. As before,h j i we can use the eigenvectors of a hermitian operator for our basis vectors. Matrices become operators in this language, M M^ . Then the eigenvalue equation becomes ! ^ M n = λn n ; (14) j i j i where the λn are real and we can take the n kets to be orthonormal: m n = δmn. Then we can write j i h j i M^ V = M^ V n = V M^ n = V λ n : (15) X n X n X n n j i n j i n j i n j i 3 Dirac notation for quantum mechanics Functions can be considered to be vectors in an infinite dimensional space, provided that they are normalizable. In quantum mechanics, wave functions can be thought of as vectors in this space. We will denote a quantum state as . This state is normalized if we make it have unit norm: = 1. j i Measurable quantities,h j i such as position, momentum, energy, angular momentum, spin, etc are all associated with operators which can act on . If O^ is an operator corresponding to some measurable quantity, its expectation valuejisi given by O^ = O^ . Note that since h i h j j i A O^ B ∗ = B O^y A ; h j j i h j j i (check this in the case of finite length vectors and matrices!) it follows that if your mea- surable quantity is real (and they always are!) then O^ = O^ ∗ implies that h i h i O^ = O^y ; h j j i h j j i or that O^ = O^y. Conclusion: Measurable quantities are associated with hermitian operators. In order to compute expectation values for given quantum states, it is often useful to choose a convenient basis. I will discuss three common bases that are often used: the position eigenstate basis, the momentum eigenstate basis, and the energy eigenstate basis. 3.1 Position eigenstate basis: x j i The position operatorx ^ =x ^y is a hermitian operator, and we can use its eigenvectors as an orthonormal basis. The state x is defined to be the eigenstate ofx ^ with eigenvalue x: j i x^ x = x x : (16) j i j i What is new here is that the eigenvalues x are not discrete, and so we use the Dirac δ-function for normalization: x x0 = δ(x x0) (orthonormality): (17) h j i − 3 The states x form a complete basis for our space and j i Z dx x x = 1^ (completeness) (18) j ih j Note that the sum over discrete basis vectors in eq. (12) has been replaced by an integral. Also, the unit operator 1^ has replaced the unit matrix I. Therefore we can expand our state in terms of the x basis vectors: j i j i = 1^ = Z dx x x = Z dx (x) x ; (19) j i j i j ih j i j i where we have defined (x) x . This (x) is nothing but our familiar wavefunction. In the present language, (x≡) are h j thei coordinates of the our state in the x basis. Note that in eq. (19) we inserted the unit operator in the guisej ofi a integralj i over x x | this technique is very powerful, and is called \inserting a complete set of states"j ih .j If we have normalized so that = 1, it follows that j i h j i 1 = = 1^ = Z dx x x = Z dx ∗(x) (x) : (20) h j i h j j i h j ih j i This is the usual normalization condition we have seen. Computing x^n for our state is particularly easy in the x basis, since x is an eigen- state of the operatorh i x ^n: j i j i x^n = x^n h i h j j i n = Z dx x^ x x h j j ih j i n = Z dx x x x h j ih j i n = Z dx x ∗(x) (x) : (21) In the next section I will discuss measuring p^ , using the position eigenstate basis. h i 3.2 Momentum eigenstate basis: p j i Another useful basis is formed by the eigenstates of the momentum operatorp ^: p^ p = p p ; p p0 = δ(p p0) ; Z dp p p = 1^ : (22) j i j i h j i − j ih j We can expand in the momentum basis as j i = Z dp p p = Z dp φ(p) p ; φ(p) p : (23) j i j ih j i j i ≡ h j i We see that φ(p) is just the wavefunction in the momentum basis. As in the above section, we can easily compute expectation values such as p^n using this basis. h i 4 It is interesting to ask how we can translate between the x and the p bases. For this, we need to know the quantity x p . We can get this by knowingj i that j i(x) and φ(p) are Fourier transforms of each other:h j i dp ipx=¯h (x) = Z φ(p)e : (24) p2πh¯ We can rewrite this as dp ipx=¯h (x) = x = Z p e : (25) h j i p2πh¯ h j i Inserting a complete set of states on the left hand side of the above equation we get dp ipx=¯h Z dp x p p = Z p e ; (26) h j ih j i p2πh¯ h j i implying that eipx=¯h x p = : (27) h j i p2πh¯ Using this result we can also compute (here y is an eigenvector ofx ^ with eigenvalue y): j i y p^ x = Z dp y p^ p p x h j j i h j j ih j i = Z dp p y p p x h j ih j i p ip(y x)=¯h = Z dp e − 2πh¯ @ dp ip(y x)=¯h = i Z e − @x 2π @ = ih¯ δ(x y) : (28) @x − Therefore if we know (x) but not φ(p), we can still compute p^ as h i p^ = p^ = Z dy Z dx y y p^ x x h i h j j i h j ih j j ih j i @ = Z dy Z dx ∗(y) ih¯ δ(x y)! (x) : (29) @x − Integrating by parts with respect to x (ignoring the boundary terms at x = , which vanish) we get ±∞ @ p^ = = Z dy Z dx ∗(y)δ(x y) ih¯ ! (x) h i − − @x @ = Z dx ∗(x) ih¯ ! (x) (30) − @x So we see that in the x representation,p ^ ih¯ @ : ! − @x 5 3.3 Energy eigenstates: n j i Finally, another operator of interest is the Hamiltonian H^ which gives the energy.
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