Introduction to Orthogonal functions:
The Orthogonal functions play a significant role in the Quantum mechanics. This is because as they afford us a set of functions, which don't mix, just the way you could resolve a vector in two dimensions in the x and y directions, correspondingly, by the unit vectors i and j. The dot product of the two unit vectors gives you zero. We would as well like to resolve our vectors in some directions. Therefore, you require knowing regarding orthogonal and Orthonormality functions. The orthonormal functions would make the possible states you can determine a system. You know such states must not mix.
Definitions:
1) We state v1 and v2 in a vector space 'V' are orthogonal if their inner product is zero, that is, (v1, v2) = 0.
2) Assume that there exists a linearly independent set {Φi}i=1 n , that is, {Φ1, Φ2, ...., Φn} , in such a way that (Φi, Φj) = 0, i ≠ j, then, {Φi}i=1 n is the orthogonal set.
3) If in addition to condition (2) above, (Φi, Φj) = 1 then, {Φi}i=1 n is the orthonormal set.
For an orthonormal set, thus, we can write (Φi, Φj) = δij, where δij is the Kronecker delta, equivalent to 0 if i ≠ j and equivalent to 1 if i = j.
As we are familiar that, if any vector in the vector space, 'V', can be written as the linear combination
v = a1Φ1 + a2Φ2 + .... + anΦn = i=1Σn aiΦi
Then we state that the space is spanned via the complete orthonormal basis {Φi}i=1 n, where (Φm, Φn) = δmn
When {Φi}i=1 n is the orthonormal set, this follows that we can recover the coefficient of expansion as shown:
(Φj, v) = (Φj, i=1Σn aiΦi) = i=1Σn ai (Φj,Φi) = aj
Furthermore,
(v,v) = (k=1Σn akΦk, i=1Σn aiΦi) = k=1Σn ak * i=1Σn ai (Φj,Φi) = i=1Σn |ai|2
If, in addition, the vector 'v' is normalized, then
i=1Σn |ai|2 = 1
Bra and Ket (Dirac) Notation:
We have written that the inner product is in the form (.,.).We could as well represent it in the form of a bra, |.>, and a ket, <.|. This is the Dirac notation. Placing the bra and the ket altogether forms a 'bracket' <.|.>. The set of vectors {Φj}j=1 n can be observed as a set of bra vectors (that is, space of vectors) {|Φj>}j=1 n. Then, we would require a dual set of vectors (that is, dual space of vectors) {<Φj|}j=1 n to be capable to write the inner product.
This follows from the foregoing, that we can represent the expansion of a wave-function:
ψ = Σj cjΦj as ψ = j=1Σn cj|Φj>
Additionally, (Φj,aΦj) = a(Φj,Φj) and (aΦj,Φj) = a*(Φj,Φj). It follows that a(Φj,Φj) = (a*Φj,Φj) = (a*)*(Φj,Φj). We can take out the given rule from this:
(Φj,aΦj) = (a*Φj,Φj)
More commonly, a could be an operator A. Then,
(Φj, AΦj) = (A+ Φj,Φj)
We can represent this in the form of:
< Φj|A|Φj > = < A+ Φj|Φj>
The above two equations become,
< Φj, v >=< Φj |i=1Σn ai|Φi > = i=1Σn ai < Φj|Φi > = aj
<v|v >=< k=1Σn akΦk | i=1Σn aiΦi > = k=1Σn ak * i=1Σn ai < Φj|Φi> = i=1Σn |ai|2
Orthogonal Functions:
An even function is symmetrical about the y-axis. In another words, a plane mirror positioned on the axis will generate an image which is precisely the function across the axis. An illustration is represented in the first part of the figure above. An odd function will require to be mirrored two times, once all along the y-axis, and once all along the x-axis to accomplish the similar effect. Second part of the figure above is an illustration of an odd function.
A function f(x) of x is stated to be an odd function when f(-x) = f(x), example: sin x, x2n+1, and a function f(x) of x is stated to be an even function when f(-x) = f(x), e.g., cos x, x2n where n = 0, 1, 2, .......
Some of the real-valued functions are odd, some are even and the rest are neither odd nor even. Though, we can write any real-valued function as the sum of an odd and an even function.
Assume that the function is h(x), and then we can write:
h(x) = f(x) + g(x)
Here f(x) is odd and g(x) is even. Then, f(-x) = -f(x) and g(-x) = g(x)
h(-x) = f(-x) + g(-x) = -f(x) + g(x)
Adding both the equation above, we get:
h(x) + h(-x) = 2g(x)
On subtracting the equations, we get:
h(x) - h(-x) = 2f(x)
It follows, thus, that
f(x) = [h(x) - h (-x)]/2
And g(x) = [h(x) + h(-x)]/2
Gram-Schmidt Orthogonalisation Procedure:
This gives a process of constructing an orthogonal set from a given set. Normalizing each and every member of the set then gives an orthonormal set. The process entails setting up the first vector, and then constructing the subsequent member of the orthogonal set by making it orthogonal to the first member of the set under construction. Then the next member of the set is made in a way to be orthogonal to the two preceding members. This method can be continued till the last member of the set is constructed.
Some useful Mathematics on Matrices:
You shall require the following as we frequently represent an operator in quantum mechanics through a matrix. We shall take as the usual basis in 3-dimensional space, {e1, e2, e3}. You might as well see this basis as {i, j, k}.
Orthogonal Matrices:
A tensor Q such that (Qa).(Qb) = a.b ∀ a,b ∈ E is known as the orthogonal matrix.
As (Qa).(Qb) = b. {QT (Qa)} = b .{(QTQ)a}, an essential and sufficient condition for Q to be orthogonal is:
QQT = I
Or equally,
Q-1 = QT
Note that:
det (QQT) = det (Q) det (QT)
det (QQT) = det (Q) det (Q)
det (QQT) = (det (Q))2 = 1
=> det (Q) = ±1
'Q' is stated to be a proper orthogonal matrix if det (Q) = 1 and an improper orthogonal matrix when det (Q) = -1
When det (Q) = 1, then
det (Q - 1) = det (Q - I) det (QT)
det (Q - 1) = det (QQT - QT) (det (A) det (B) = det (AB) for any two square matrices)
det (Q - 1) = det (I - QT) (QQT = I for an orthogonal matrix Q)
det (Q - 1) = det (IT - QTT) (det A = det AT for any square matrix A.)
det (Q - 1) = +det (I - Q) (IT = I and QTT = I)
det (Q - 1) = (det (-A) = - det (A) for any square matrix A.)
det (Q - 1) = 0 (if a number is equivalent to its negative, it should be zero)
Thus, 1 is an Eigen value in such a way that ∃ e3 ∋Qe3 = e3
Symmetric Matrices:
For a symmetric matrix A, A = AT
Select e1, e2, e3 as Eigen-vectors of A having Eigen values λ1, λ2, λ3.
Aek = λkek
λk (ek . ej) = Aek . ej
λk (ek . ej) = ek . ATej
λk (ek . ej) = ek . Aej
λk (ek . ej) = λ(ek . ej)
This signifies that if λj ≠ λk, then ei . ej = δij
This signifies that we could stand for a symmetric matrix as a diagonal matrix with only the entries Aii = λi:
This result is termed to as the spectral representation of a symmetric matrix.
Hermitian Matrices:
The Adjoint (or Hermitian conjugate) of a matrix A is represented by:
Adj (A) = A+ = ((A)T)*
The Hermitian matrix is the complex equivalent of the real symmetric matrix, satisfying
A+ = A
Unitary Matrices:
The complex analogue of the real orthogonal matrix is a unitary matrix, that is, AA+ = I or equally,
A+ = A-1
Normal Matrices"
A normal matrix is one which commutes by its Hermitian conjugate.
That is,
AA+ = A+A
Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)
Expand your confidence, grow study skills and improve your grades.
Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.
Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.
Free to know our price and packages for online physics tutoring. Chat with us or submit request at [email protected]
tutorsglobe.com mechanism of ascent of sap assignment help-homework help by online water transport tutors
tutorsglobe.com breeding experiments assignment help-homework help by online biology in human welfare tutors
The apt Information Technology Assignment Help tutors are available 24x7 to resolve all your worries at low prices and fetch you A++ grades
tutorsglobe.com streptococcus pyogenes assignment help-homework help by online medical bacteriology tutors
the meaning of television is “to see from a distance” .the 1st illustration of actual television was provided through j.l.baird in uk (united kingdom) and c.f.jenkins in us (united states) around 1927.
tutorsglobe.com biological database assignment help-homework help by online modern genetics tutors
Theory and lecture notes of Transaction scheduling all along with the key concepts of transaction scheduling, transaction management, Primed transactions. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Transaction scheduling.
tutorsglobe.com properties of copper sulphate assignment help-homework help by online copper sulphate tutors
Opt the fastest way to earn top grades with Political Geography Assignment Help at the most feasible prices!
tutorsglobe.com albinism assignment help-homework help by online inborn errors of metabolism tutors
Theory and lecture notes of Theory of Common Mode Rejection Ratio II, all along with the key concepts of Mismatch in Gain Determining Resistors, Finite CMRR, Operational Amplifier. Tutorsglobe offers homework help, assignment help and tutor’s assistance on Theory of Common Mode Rejection Ratio II.
What is process costing - The system of process costing is appropriate for industries including continuous production of similar product or products via the same process or set of processes.
Want an ideal American Politics Assignment Help to resolve tough tasks at Apt Price? Hire us to secure notable grades!
www.tutorsglobe.com offers theory of unemployment & types of unemployment, economics assignment help - homework help and writing assignments.
TutorsGlobe.com Alkanoates Assignment Help-Homework Help by Online Access Chemistry Tutors
1944017
Questions Asked
3689
Tutors
1492642
Questions Answered
Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!