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BRAS AND KETS

In Quantum Mechanics , the state of a Physical system is identified with a Vector in a Complex Hilbert Space , ''H''. Each Vector is called a "ket", and written as

  Every Ket <math>\psi Angle</math> Has A "http://wwwinformationdelightinfo/encyclopedia/entry/Duality_(mathematics)" class="copylinks">Dual bra, written as
  :<math>\langle\psi Ho Angle \bigg( \psi angle \,\ ho angle \bigg)</math> for all kets <math> ho angle</math>
  Given Any Bra <math>\langle\phi</math>, Kets <math>\psi 1 Angle</math> And <math>\psi 2 Angle</math>, And "http://wwwinformationdelightinfo/encyclopedia/entry/complex_number" class="copylinks">Complex Number s ''c''<sub>1</sub> and ''c''<sub>2</sub>, then, since bras are ''linear'' functionals,
  ::<math>\langle\phi \ \bigg( C 1\psi 1 Angle + C 2\psi 2 Angle \bigg) c_1\langle\phi\psi_1 angle + c_2\langle\phi\psi_2 angle </math>
  ::<math>\bigg(c 1 \langle\phi 1 + C 2 \langle\phi 2\bigg) \ \psi Angle c_1 \langle\phi_1\psi angle + c_2\langle\phi_2\psi angle </math>
  Given Any Kets <math>\psi 1 Angle</math> And <math>\psi 2 Angle</math>, And "http://wwwinformationdelightinfo/encyclopedia/entry/complex_number" class="copylinks">Complex Number s ''c''<sub>1</sub> and ''c''<sub>2</sub>, from the properties of the inner product (with c denoting the Complex Conjugate of c),
  ::<math>\langle\phi\psi Angle \langle\psi\phi angle^</math>
  If ''A'' : ''H'' &rarr ''H'' Is A "http://wwwinformationdelightinfo/encyclopedia/entry/linear_operator" class="copylinks">Linear Operator , we can apply ''A'' to the ket <math>\psi angle</math> to obtain the ket <math>(A\psi angle)</math> Linear operators are ubiquitous in the theory of quantum mechanics For example, Hermitian Operators are used to represent observable physical quantities, such as Energy or Momentum , whereas Unitary linear operators represent transformative processes such as rotation or the progression of time
  :<math>\bigg(\langle\phiA\bigg) \ \psi Angle \langle\phi \ \bigg(A\psi angle\bigg)</math>
  A Convenient Way To Define Linear Operators On ''H'' Is Given By The "http://wwwinformationdelightinfo/encyclopedia/entry/outer_product" class="copylinks">Outer Product : if <math>\langle\phi</math> is a bra and <math>\psi angle</math> is a ket, the outer product
  Denotes The Rank One Operator That Maps The Ket <math> Ho Angle</math> To The Ket <math>\phi Angle\langle\psi Ho Angle</math> (where <math>\langle\psi Ho Angle</math> Is A Scalar Multiplying The Vector <math>\phi Angle</math>) One Of The Uses Of The Outer Product Is To Construct "http://wwwinformationdelightinfo/encyclopedia/entry/projection_operator" class="copylinks">Projection Operator s Given a ket <math>\psi angle</math> of norm 1, the orthogonal projection onto the Subspace spanned by <math>\psi angle</math> is
  For Instance, The Hilbert Space Of A "http://wwwinformationdelightinfo/encyclopedia/entry/spin_(physics)" class="copylinks">Zero-spin point particle is spanned by a position basis <math>\lbrace\mathbf{x} angle brace</math>, where the label '''x''' extends over the set of position vectors Starting from any ket <math>\psi angle</math> in this Hilbert space, we can ''define'' a complex scalar function of '''x''', known as a Wavefunction :
  :<math>\mathbf{p} \psi(\mathbf{x}) \equiv \lang \mathbf{x} \mathbf{p}\psi Ang - i \hbar