6_postulates2-2

p a m dirac response to dirac look for working

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Unformatted text preview: soluble." P. A. M. Dirac Response to Dirac: Look for working approximations like Variational Theory and Perturbation Theory (Ch. 7) 2 9/16/2013 Postulate 1 The state of a quantum-mechanical system is completely specified by a function ( x ) that depends upon the coordinate of the particle. All possible information about the system can be derived from ( x ) . This function, called the wavefunction or the state function, has the important property that * ( x) ( x) dx is the probability that the particle lies in the interval dx , located at the position x . can be complex 1-D probability * ( x ) ( x) dx * 3-D probability ( x, y, z ) ( x, y, z )dxdydz d 1 * Normalization must be normalizable (quadratically integrable) and result unitless d must be single valued functions and finite dx both ( x ) and Postulate 1 Ψ∗ , Ψ, Ψ∗ one dimensional two dimensional three dimensional 1 Ψ∗ Ψ∗ , Ψ, ,, Ψ∗ , 1 Ψ ,, ,,, Ψ, is a unitless probability units of Ψ: 1/ 1 units of Ψ: 1/ Ψ ,,, 1 units of Ψ: 1/ Probability that particle is in volume element d dxdydz Ψ∗Ψ Ψ∗ ,,, is Ψ ,,, If wavefunction is a stationary state (solution of the time independent Schrödinger equation) Ψ∗Ψ ,, ,, / ,, / ,, Just normalize the spatial part unit probability of finding the particle in all space, so Ψ∗Ψ ∗ 1 Triple integration 3 9/16/2013 Normalization If the normalization integration doesn’t equal one, then the function is not normalized ∗ To normalize, divide the wavefunction by /. i.e. Check. , ∗ ∗ 1 Bra-Ket Notation or Dirac Notation ∗ | Bra Ket Bra is the complex conjugate of a Ket Postulate 2 To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics. ˆ ˆ ˆ Ac1 f1 ( x ) c2 f 2 ( x ) c1 Af1 ( x ) c2 Af 2 ( x ) ∗ ∗ flipped and starred: flipped (g and f are exchanged) and starred (take complex conjugate of result); transposed and complex conjugated ∗ ∗ Hermitian operator also works in the opposite direction ? What if Expression is expectation value, like Postulate 4 ∗ ∗ ∗ expression equals same expression starred – only true if real • guarantees that eigenvalues of Postulate 4 will be real • mathematically: the operators must be self adjoint, • Hermitian operators have complete sets of orthonormal eigenfunctions Postulate 4 in Dirac Braket Notation ∗ Bra Ket | Hermitian in Dirac Bra...
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