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** 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 v_{1} and v_{2} in a vector space 'V' are orthogonal if their inner product is zero, that is, (v_{1}, v_{2}) = 0.

2) Assume that there exists a linearly independent set {Φ_{i}}_{i=1} ^{n} , that is, {Φ_{1}, Φ_{2}, ...., Φ_{n}} , in such a way that (Φ_{i}, Φ_{j}) = 0, i ≠ j, then, {Φ_{i}}_{i=1} ^{n} is the orthogonal set.

3) If in addition to condition (2) above, (Φ_{i}, Φ_{j}) = 1 then, {Φ_{i}}_{i=1} ^{n }is the orthonormal set.

For an orthonormal set, thus, we can write (Φ_{i}, Φ_{j}) = δ_{ij}, where δ_{ij} 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 = a_{1}Φ_{1} + a_{2}Φ_{2} + .... + a_{n}Φ_{n} = _{i=1}Σ^{n} a_{i}Φ_{i}

Then we state that the space is spanned via the complete orthonormal basis {Φ_{i}}_{i=1} ^{n}, where (Φ_{m}, Φ_{n}) = δ_{mn}

When {Φ_{i}}_{i=1} ^{n} is the orthonormal set, this follows that we can recover the coefficient of expansion as shown:

(Φ_{j}, v) = (Φ_{j}, _{i=1}Σ^{n} a_{i}Φ_{i}) = _{i=1}Σ^{n} a_{i} (Φ_{j},Φ_{i}) = a_{j}

Furthermore,

(v,v) = (_{k=1}Σ^{n} a_{k}Φ_{k}, _{i=1}Σ^{n} a_{i}Φ_{i}) = _{k=1}Σ^{n} a_{k} * _{i=1}Σ^{n} a_{i} (Φ_{j},Φ_{i}) = _{i=1}Σ^{n} |a_{i}|^{2}

If, in addition, the vector 'v' is normalized, then

_{i=1}Σ^{n} |a_{i}|^{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 {Φ_{j}}_{j=1} ^{n} can be observed as a set of bra vectors (that is, space of vectors) {|Φ_{j}>}_{j=1} ^{n}. Then, we would require a dual set of vectors (that is, dual space of vectors) {<Φ_{j}|}_{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 = Σ_{j} c_{j}Φ_{j} as ψ = _{j=1}Σ^{n} c_{j}|Φ_{j}>

Additionally, (Φ_{j},aΦ_{j}) = a(Φ_{j},Φ_{j}) and (aΦ_{j},Φ_{j}) = a*(Φ_{j},Φ_{j}). It follows that a(Φ_{j},Φ_{j}) = (a*Φ_{j},Φ_{j}) = (a*)*(Φ_{j},Φ_{j}). We can take out the given rule from this:

(Φ_{j},aΦ_{j}) = (a*Φ_{j},Φ_{j})

More commonly, a could be an operator A. Then,

(Φ_{j}, AΦ_{j}) = (A^{+} Φ_{j},Φ_{j})

We can represent this in the form of:

< Φ_{j}|A|Φ_{j} > = < A^{+} Φ_{j}|Φ_{j}>

The above two equations become,

< Φ_{j}, v >=< Φ_{j} |_{i=1}Σ^{n} a_{i}|Φ_{i} > = _{i=1}Σ^{n} a_{i} < Φ_{j}|Φ_{i} > = a_{j}

<v|v >=< _{k=1}Σ^{n }a_{k}Φ_{k} | _{i=1}Σ^{n} a_{i}Φ_{i }> = _{k=1}Σ^{n} a_{k} * _{i=1}Σ^{n} a_{i} < Φ_{j}|Φ_{i}> = _{i=1}Σ^{n }|a_{i}|^{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, x^{2n+1}, and a function f(x) of x is stated to be an even function when f(-x) = f(x), e.g., cos x, x^{2n} 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, {e_{1}, e_{2}, e_{3}}. 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. {Q^{T} (Qa)} = b .{(Q^{T}Q)a}, an essential and sufficient condition for Q to be orthogonal is:

QQ^{T} = I

Or equally,

Q^{-1 }= Q^{T}

Note that:

det (QQ^{T}) = det (Q) det (Q^{T})

det (QQ^{T}) = det (Q) det (Q)

det (QQ^{T}) = (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 (Q^{T})

det (Q - 1) = det (QQ^{T} - Q^{T}) (det (A) det (B) = det (AB) for any two square matrices)

det (Q - 1) = det (I - Q^{T}) (QQ^{T} = I for an orthogonal matrix Q)

det (Q - 1) = det (I^{T} - Q^{TT}) (det A = det AT for any square matrix A.)

det (Q - 1) = +det (I - Q) (I^{T} = I and Q^{TT} = 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 ∃ e_{3} ∋Qe_{3 }= e_{3}

Symmetric Matrices:

For a symmetric matrix A, A = A^{T}

Select e_{1}, e_{2}, e_{3} as Eigen-vectors of A having Eigen values λ_{1}, λ_{2}, λ_{3}.

Ae_{k} = λ_{k}e_{k}

λ_{k} (e_{k} . e_{j}) = Ae_{k} . e_{j}

λ_{k} (e_{k} . e_{j}) = e_{k} . A^{T}e_{j}

λ_{k} (e_{k} . e_{j}) = e_{k} . Ae_{j}

λ_{k} (e_{k} . e_{j}) = λ(e_{k} . _{ej})

This signifies that if λ_{j} ≠ λ_{k}, then e_{i} . e_{j} = δ_{ij}

This signifies that we could stand for a symmetric matrix as a diagonal matrix with only the entries A_{ii }= λ_{i}:

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

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