1 Key points Lecture 6: Vector Spaces II - Matrix Representations Linear Operators Matrix representation of vectors and operators Hermite conjugate (adjoint) operator Hermitian operator (self-adjoint) operator Projection operator Maple commands Transpose HermitianTranspose 2 Matrix representation of vector Consider a vector expanded with a basis set, You can specify the vector by a set of coefficients and a basis set If you are using the samea basis set to decribe many vectors, it is redundant to specify the basis set each time You need to show only the coefficients A common way to express a vector in a basis set is a column matrix Examples 3-d Euclidean space: a=3e x -2e y +4e z Spin 1/2: = 1 Even a function of continuous variable can be expressed in a descrete matrix form For
where f(x) Examples Fourier sin expansion of an odd function: Its dual (adjoint) is a Hermitian conjugate (transpose+complex conjugate) Then, the inner product follows the standar matrix operation Exercise: Electron spin Consider two spin states = matrix form and show that they are orthogonal to each other Answer
= = Using Maple command, = 0 By direct calculation, = 0 Using Maple short cuts, = 0 Note that Maple automatically use Hermite conjugate in DotProduct 3 Linear Operators An operator transforms one vector to another where is Hermite conjugate (adjoint) of I is identity operator: I Exercise Show that Answer Let Then, and By definition, Thus, Example: Rotation
which can be expressed as where and To rotate a row vector which can be expressed as Exercise Rotate a vector on the x axis by about z axis Answer Construct a rotation matrix (1) Define an initial vector
(2) Rotate it (3) Example: Harmonic oscillator The Newton equation harmonic oscillator is which can be written with two first order differential equations and These two equations can be expressed in a matric form This means a vector is transformed to by a linear operator Example: Coupled oscillators Consider two identical particle of mass equations of motion are connected by a spring (spring constant ) The which can be written in a matrix form The position
is transformed to acceleration by a linear operator Example: Poisson equation An electrostatic potential induced by an electric charge density is determined by Poisson equation (Laplacian or Laplace operator) by a linear operator Example: Schrödinger equation The fundamental equation of motion for quantum particles is Schrödinger equation A state vector is transformed to by a linear operator (Hamiltonian) Dyad where (a complex number) Exercise: Hermite conjugate of dyad Show that if + = Answer Applying the operator on an arbitoray vector, we obtain a new vector where Its dual is Hence, 4 Matrix representation of operators Using a basis set, = = Multiplying Cu i, = = where is a matrix representation of the operator is equivalent to i =
Examples, U^ Identity operator is a diagonal matrix whose diagonal elements are all 1 Exercise: Flipping spins An operator F flips the spin of electron: and, where and 5 Adjoint (Hermit conjugate) and Hermitian Operator The matrix representation of is
which means In the matrix representation is complex conjugate and transpose of When =, the operator A is Hermitian (self adjoint) A general relation Ca O O + * is equivalent to (ABC) + =C + B + A + Example (4) (5) (6) = = (7) (8) Hence
Example An operator For functions and, is a Hermitian (self-adjoint) operator = = where we used bundary conditions and On the other hand, Hence, = Exercise Show that the adjoint of operator is Answer Let = = where we used bundary conditions and On the other hand,
Hence, = 6 Projection Operator Projection operator is any operator that is idempotent: In physics, projection operators is often expressed as where is a normalized vector In particular, projection onto a base vector is done by P= u i u i Exercise Show that is idempotent The projection operator projects any vector on to a direction of In particular, projection onto a base vector is done by i = u i u i c Example 1 u 1 2 u 2 N u N 2 c 2 u 2 Consider a vector a=3e x -5e y +2e z We want to project this vector onto a line specified by another vector s=6e x +2e y +1e z Define s in matrix form (9) Normalize s, (10)
(11) Construct the projection operator (12) Define a in matrix form: (13) Project to (14) Another way to calculate it (15)
(16) a s b = l s Example Project to where where is a complex number
g f h = l f 7 Completeness (Clausure Relation) Example In 2-dimentional vector space, we have orthonormal vectors u 1 and u 2 Show the completeness relation Define the base vectors
Construct projection operator 1 = u 1 u 1 and 2 = u 2 u 2 (17) You can use Maple shortcut, (18) Sum of the projection operators should be I = 8 Orthogonal and Unitary Transformations Consider operators that preserve the norm of vectors: we have +, + + Any operator U is saied to be unitary and U is a unitary operator When U is a real operator, it is said to be orthogonal Noting the definition of inverse U -1 + = -1 Example Consider the rotational matrix
(20) simplify = (22) (23) Therefore, Exercise Construct an operator that transforms u 1
and u 2 to w 1 and w 2, respectively Show that the operator is unitary Answer (24)
(25) (26) = Hence the operator is a unitary operator Homework: Due on 9/19, 11am 61 There are three potential wells,, and When a quantum particle is trapped in a potential, its state vector is given by If the particle is found to be in all three potentials with equal probability, what is the state vector of the particle The given information is not sufficient to determine a precise vector Answer a most general expression that is consistent with the given condition 62 A force is given by a vector ^ Find the component of the vector in the direction specified by a vector ^ Note that is not normalized 63 Consider a function whre is a positive constant 1 Project the above function on to another function 2 Project the above function on to a function 64 Find the Hermite conjugate of the following operators
Is this operator self adjoint (Hermitian)? 65 In quantum mechanics, the kinetic energy operator is given by Show that is self adjoint (Hermitian) operator