1 Infinite-Dimensional Vector Spaces

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1 Theoretical Physics Notes 4: Linear Operators In this installment of the notes, we move from linear operators in a finitedimensional vector space (which can be represented as matrices) to linear operators in an infinite-dimensional vector space (which cannot be written as matrices). The utility of Dirac notation becomes evident here: unlike matrices, it can be used when the vector space is infinite (either with a discrete index like finite matrices have, or with a continuous one). 1 Infinite-Dimensional Vector Spaces Just like matrix multiplication is a linear operation on a finite-dimensional vector space, taking a derivative is also a linear operation: the derivative d/dx of a linear combination of functions αf(x)+βg(x) is just αdf/dx+βdg(x)/dx, if α and β are constants. Vector spaces of functions are infinite-dimensional, however, and so we can no longer use matrix notation (but we can and will continue to use Dirac notation). We can specify a function of x by simply giving its value at every point x. We can formalize this by introducing the basis vectors { x }, where there is one such vector for each point in the domain. It is convenient to think of these as being the eigenvectors of a position operator ˆX, so that for any specific point x 0 we will have ˆX x 0 = x 0 x 0. Since x is a continuous variable on the real line, we cannot run over all its values by using a summation symbol; instead, we replace summations by an integration over the continuous variable. We then express orthonormality by a Dirac delta function rather than a Kronecker delta symbol: for instance, orthonormality of the kets which are eigenkets of the pointlike position measurement operator is expressed as x x = δ(x x). Just like a Kronecker delta symbol produces 1 when summed over all values of a discrete index, a Dirac delta produces 1 when integrated over all values of a continuous index. Similarly, just like in the finite-dimensional case, completeness is written as an outer product of basis vectors and their canonically orthonormal dual basis vectors, using the same label for each and spanning over all possible labels. When the label is continuous, this becomes an integral: we write the identity operator Î = dx x x. This allows us to write a function, thought of as a ket vector in our space, by specifying its values at all positions using completeness: f = Î f = dx x x f. Much of this is familiar notation: the ordinary numbers x f = f(x) are here to be thought of the value of f at a specific point x though, rather than as the entire abstract function along the whole region where it is defined. The integration dx collects together all such values and so gives us the entire function, and the basis vectors x turn the whole expression into a ket vector (ready to be acted upon by a covector, or a linear operator). 1

2 1.1 Fourier Space We don t usually bother to write the basis vectors such as x, so why all the fuss and extra notation? One reason is that so we can seamlessly change the basis being used. For instance, for piecewise continuous functions it is an equally complete amount of information to give their Fourier transform. Instead of using the position basis x, we can use the wavenumber basis k: orthonormality becomes k 1 k 2 = δ(k 1 k 2 ), and completeness becomes dk k k = Î. The same ket function f can be expressed in Fourier space: f = Î f = dk k k f = dk k f(k). The abstract ket vector f need not have either basis specified in manipulations. Putting in completeness of x and completeness of k together in one expression shows us how to take the Fourier transform and how to take the inverse transform. Sometimes, an infinite-dimensional vector space can be specified by either a continuous index or a discrete index! Turning continuous indices into discrete ones is often loosely called quantization (which is really a misuse of the term it s more correctly used to mean applying the rules of quantum mechanics to a classical system, which does not always turn continuous indices into discrete ones). Quantization of parameters which label functions occurs when some definite boundary conditions are imposed upon the functions. A simple example is found in the functions which are kernels of the Fourier transform and the inverse Fourier transform, namely k x = N exp( ikx) and x k = N exp(ikx), respectively, where N is a normalization constant. If we move to the circle so that x becomes φ, we must impose periodic boundary conditions on the functions: exp( ikφ) must equal exp[ ik(φ + 2π)] for every φ. Thus exp[ ik(2π)] = 1, and so k becomes restricted to integer values (n) only, and so Fourier transforms become Fourier series. Similarly, an infintely long vibrating string can wiggle with any wavelengths, but if two points are clamped so they cannot vibrate, only discrete wavelengths are allowed between the points. On the real line we have the completeness measure in Fourier space, which is usuallydefinedas dk k k = Î. Thenormalizationconstantscanbechosenas therealnumbersn = N = 1 2π forthe k x, andwegetthefourierrepresentation of the Dirac delta function: x x = (2π) 1 dkexp[ik(x x )] = δ(x x ). On the circle, the same normalization constants work but our completeness relationinsteadlookslikeaninfinitesum: δ(φ φ ) = (2π) 1 m= exp[im(φ φ )]. Orthogonality of the Fourier modes is a Kronecker delta symbol instead of a delta function: m n = δ mn. Inserting completeness of the coordinate angle φ gives this as the integral (2π) 1 2π 0 dφexp[iφ(m n)] = δ mn, as is easy to check by direct integration. Functions on the circle can either be given in their continuous (φ) basis, or the discrete (m) basis, often called the angular momentum basis. In general, the same normalization constants (such as (2π) 1 above) which appear in the orthogonality relations also appear in the completeness relations. Different choices of these constants will naturally lead to different constants 2

3 appearing in formulas. For example, it is common to use momentum p rather than wavevector k as a variable, where p = k. The correct transition functions between position and momentum space then pick up an extra constant: x p = (1/ 2π )exp(ipx/ ). Note that this is different than simply substituting k = p/ in the corresponding wavevector formula x k = (1/ 2π)exp(ikx) by having an extra in the denominator. This allows us to write both dk k k = Î and dp p p = Î; the extra in the denominator is required for consistency to account for the fact that dk = dp/. 2 Differential Operators One of the main purposes for going to Fourier space is that derivative operators are diagonal in Fourier space: d/dx acting on e ikx simply returns an eigenvalue (ik) times the original function e ikx. In position space, this is definitely not true: a derivative acting on a general function f(x) returns f (x) which is generally a completely different function (and not just a number times the original function). This allows us to turn differential equations (involving d/dx) into algebraic ones (replacing d/dx by ik everywhere it appears) after transforming to Fourier space. We then solve the algebraic equation in Fourier space, and then transform back to position (x) space, to give the solution to the original problem. 2.1 Some Important Operations on Linear Operators Differential operators are linear, as mentioned at the beginning of these notes. We can manipulate linear operators to give other operators that are related to the original operator and which are useful in solving specific problems, in much the same fashion that we could manipulate matrices to give solutions of matrix problems (such as finding the matrix inverse, to solve a system of linear equations). Here we list a few of the important ones Powers Positive integral powers of a differential operator are simple to obtain in terms of the original operator: to square the operator, you simply apply it twice (e.g. (d/dx) 2 is just d/dx once and then again). They are again differential operators. Mathematically they are very nice in the sense that they are local, meaning we only need to know the function s values in the neighborhood of some point to know its derivatives at that point. In contrast, negative integral powers of a differential operator are nonlocal: one usually needs to know the value of the function at remote points as well as local points to construct them. This should make perfect sense: to undo a derivative, we have to do an integral! The value of the integral of a function depends upon all values of the function between the endpoint where you begin the integral and the point where you want to know the value of the integral, not just on the behavior of the function nearby. 3

4 We can even formally construct fractional powers of a differential operator, by diagonalizing the operator! For instance, (d/dx) p becomes simply (ik) p after transforming to Fourier space, where p is an arbitrary number (even complex!). There is, of course, the usual difficulty with multivaluedness to contend with when doing so (e.g., (z) has two values everywhere except at z = 0). Negative and fractional powers of differential operators are usually called pseudodifferential operators by mathematicians Exponential Any analytic function of a linear operator may be defined by its Taylor series, so if we know how to do positive integral powers, we can do arbitrary analytic functions (e.g. exp, sin, cosh) of the operator as well. In practice, the most common one of these which shows up in physics equations is the exponential. A hopefully familiar example is the time-dependent Schrödinger equation: i ψ = Ĥ ψ t where Ĥ is a differential operator. If wh is independent of time, then the solution is formally given by an exponential: ψ(t) = exp( iĥt/ ) ψ(t = 0) where ψ(t) may be expressed in any basis (such as x or k), so we have suppressed that variable in the notation. (If Ĥ depends upon time, then Ĥt above needs to be replaced by Ĥdt, and we also need to define the time-ordered exponential since Ĥ may not commute with itself at different times.) The operator Û(t) exp( iĥt/ ) is extremely important in non-relativistic quantum mechanics; it is called the non-relativistic propagator. Note that it is unitary provided that Ĥ is Hermitian. For the free particle, the Hamiltonian in position space involves the Laplacian, but in k space it is simply ( k) 2 /(2m). This allows us to formally write Ĥ = dk k ( k) 2 /(2m) k (diagonal in k space!), so the propagator (in operator notation) is Û(t) = dk k exp [ i k 2 t/(2m) ] k. Written in this form, it is ready to operate on an arbitrary wavefunction ψ ; the product k ψ signifies that we will be finding ψ in Fourier space. The free propagator in position space will not be diagonal, since taking a derivative of a function gives a different function (not just a constant times the original function). To find it, we insert completeness on both sides of the equation, with a dx x x on one side, and dx x x on the other (with a different dummy variable of integration!). This gives a Gaussian integral over k which is simple to evaluate; we will do so later in the course. 4

5 2.1.3 Green Functions A Green Function of a linear operator is simply the inverse to the operator, provided that it exists (operators with eigenvalues of zero have no inverse). So it s really only a very special case of the powers defined above, but is important enough to get its own name. The Green function of a differential operator will always be used inside an integral, since inverses to derivatives are integrals. They can (and usually do) therefore have singular behavior, similar to Dirac delta functions (which are also used in integrals). An example which will take up many weeks of this course is the Green function for the Laplacian, which is used to solve Poisson s equation (the inhomogeneous version of Laplace s equation). It arises naturally in electrostatics by combining Coulomb s Law with E = Φ: 2 Φ(x) = ρ(x)/ǫ 0, where ρ is the (known) charge density in the region and Φ is the (unknown) potential (= voltage ). The solution is given by the integral Φ(x) = 1 d 3 x G(x;x )ρ(x ) ǫ 0 where the Green function G(x;x ) satisfies the equation 2 G = δ 3 (x x ), with boundary conditions on G appropriate to the boundary conditions imposed on the potential Φ in the region. In this way, we are constructing the solution to all such equations at once: one for any charge density ρ given in the region. The delta function allows us to solve the equation independently of the specific ρ given. Formally, we can write that G = Î/ 2. There are many ways to construct Green functions, and we shall explore enough of them that a separate handout on Green functions will be given later. In addition to that for the negative Laplacian, the Green function for the wave operator will also be very useful to us in this course Resolvent Closely related to the Green function of an operator Ĥ is its resolvent operator, which depends upon both the operator and an additional complex variable z. It is defined formally as R(Ĥ,z) = Î/(Ĥ zî), so for any fixed value of z it is simply the Green function of Ĥ zî. It is important for many reasons; foremost amongst these is that it will possess simple poles at locations wherever z is an isolated eigenvalue of Ĥ. So if the resolvent is known, we can use the Cauchy integral formula to construct projection operators onto the eigenspaces with any particular eigenvalues: if Ĥ n = λ n n, then P n = n n = 1 2πi R(Ĥ,z)dz where C is a counterclockwise contour which goes around z = λ n but no other eigenvalues of Ĥ (and a C which goes around many eigenvalues will produce the sum of projection operators onto each eigenspace). The negative sign is needed because we have Ĥ z rather than z Ĥ in the denominator; using instead a clockwise contour would eliminate the minus sign. C 5

6 2.1.5 Heat Kernel Another exponential operator (which is defined for elliptic differential operators with positive eigenvalues, such as minus the Laplacian) which is closely related to the propagator is the heat kernel. The heat kernel of such an operator Ĥ is formally written as K(Ĥ,t) = exp( Ĥt), where t is an auxiliary nonnegative real number (often called a Schwinger Parameter ). The heat kernel is the solution of the differential equation t K +ĤK = 0 with the initial condition (in position basis) x K(Ĥ,t = 0) x = δ(x x ). The parameter t looks formally similar to time in the heat (diffusion) equation when Ĥ is the negative Laplacian, but is really just an auxiliary parameter that may have nothing to do with actual physical time. (The restriction to Ĥ with positive eigenvalues only is so that K converges as t ; if Ĥ has any negative eigenvalues then expressions for K will generally diverge. This is the reason why we use the negative Laplacian; the positive Laplacian has negative eigenvalues.) The heat kernel can itself be used to construct other associated operators. For instance, the Green function of Ĥ is given by the integral G = K(Ĥ,t)dt. (To verify this, evaluate it using as a basis the eigenvectors 0 of Ĥ.) This is the fastest way to find the Green function of the negative Laplacian in spaces of arbitrary dimensions. We can also construct other powers of Ĥ by inserting various powers of t into the integral (if the expression converges). The resolvent and heat kernel of an operator are intimately related; the resolvent is the Laplace transform of the heat kernel (with z as the argument of the transform: 1/(Ĥ z) = exp( Ĥt+zt)dt. We will not explore the 0 resolvent and heat kernel much in this class, but they are useful enough that you should know about them. 6