**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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