1. التفسيرالرياضي لدالة الموجة وشروط قبول الدالة

2. (a) غير مقبولة (ليست أحادية القيمة).
(b) غير مقبولة (غير مستمرة).
(c) غير مقبولة (قيمة لانهائية حيث x=∞).
(d)مقبولة.

3. Molecular Structure and Spectroscopy
7/6/2018
Molecular Structure and Spectroscopy
The wave function, 
The wave function is often complex-valued.
The absolute square |y|2 = y*y is always real and positive
(y* is the complete conjugate of y).
|y|2 is proportional to the probability per unit volume of finding a particle at a given point at some instant.
The wave function contains within it all the information that can be known about the particle.
Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time.
This Schrödinger Wave Equation represents a key element in quantum mechanics.
Dr. Sabry El-Taher

4. فروض الكم

5. Postulates of Quantum Mechanics
Postulate 1. The WavefunctionPostulate:
“The state of a quantum mechanical system is described by a wave function  (x,t), of the coordinates of all the particles and of time. It is called the state function, contains all the information that can be determined about the system. It must be single-valued, continuous, and quadratically (or square) integrable.”

6. Examples of the meaning of “The coordinates of all the particles”
For a single particle moving in one dimension:
For a single particle moving in three dimensions:
For two particles moving in three dimensions:
Square-integrable means that the normalization integral is finite:

7. :postulate2:The Observables and Operators Postulate
“To every physical observable there corresponds a linear Hermitian operator in quantum mechanics”.
المؤثرات الميكانيكية الكمية المرادفة للخواص الفيزيائية خواص النظام مؤثر لا بد ان تكون خطية وهرميتية
An operator L is linear if and only if :

8. Hermitian operators: examples

10. Postulate3:The eigenfunctions and eigenvalues
“The only possible values that can result from measurements of the
Physical observable G are the eigenvalues gi of the equation.”
eigenfunction
eigenvalue
operator
The eigenfunctions are required to be well-behaved.
Example: The time-independent Schrödinger equation:
Important fact: The eigenvalues of a Hermitian operator are real.

11. Postulate 4. Basis Set of functions
“The set of eigenfunctions of Hermitian operators Q will form a complete set of linearly independent functions.”

12. Postulate 5: Expectation values
“For a system described by a given normalized wavefunction, the expectation value of a physical observable can be found by performing the expectation value integral with respect to that wavefunction.”

13. A physical system is described by the wave function Ψ, which can always be written as a linear combination of the eigenfunctions of a Hermitian operator Q:
A measure of Q for the state Ψ will give as a result any of its eigenvalues qn, each with a probability |cn|2, so that
The normalization condition of the wavefunction implies that
A measurement of Q forces the system to be in one of the eigenstates, Ψn, of Q: any subsequent measure of Q will give the result qn

14. Postulate 6: Time evolution of the system
“The time evolution of the wavefunction is given by the time dependent Schrödinger equation”
If Ψ(x,y,z; t) is the wavefunction for a physical system at an initial time and the system is free of external interactions, then the evolution in time of the wavefunction is given by
This is a linear, homogeneous differential equation, so the linear combination of any two solutions is also a solution.

16. الدالات متعامدة :
الدالات المعايرة :