Postulates of quantum mechanics

Postulate of quantum mechanics for inorganic chemistry (MSc)

1. DEPARTMENT OF CHEMISTY UNIVERSITY SCHOOL OF SCIENCE

2. “Postulates of Quantum Mechanics” Name: Parmar Nilesh D Roll No. 67 M.Sc -1 With Reference : Donald A. McQuarrie , Ira N. Levine-Quantum Chemistry

3. Postulate 1 ‘’The state of a quantum-mechanical system is described by a function Ψ (r, t) that depends on the coordinates of the article and on time’’ This function, called the wave function or state function , contains all the information that can be determined about the system. Ψ (r, t) or Ψ (x ,y ,z ,t) This function has the important property that Ψ (x ,y ,z ,t) is the probability that the particle lies in the volume element dx dy dz located at r at time t .

4. we may anticipate that same physically observable property of the electron is connected to, Ψ 2 (x ,y ,z) Or more generally , Ψ ӿ (x ,y ,z) Ψ(x ,y ,z) If Ψ ӿ is a complex wave function. For a system having electron there are two ways in which |Ψ 2 | or |Ψ ӿ Ψ| can be interpreted. Either |Ψ 2 | may be regarded as a measure of the density of electrons in a small volume dr 2 in a certain of space. The born interpretation of Ψ is that |Ψ ӿ Ψ|dr or |Ψ 2 | is probability of finding the electrons in an infinitesimal region between & and r+dr. (dr= dx dy dz )

5. The born interpretation of Ψ is that |Ψ ӿ Ψ|dr or |Ψ 2 | is probability of finding the electrons in an infinitesimal region between & and r+dr. (dr= dx dy dz ) Under such circumstances the best thing one can do is to accepted the born interpretation of the wave function. For this reason |Ψ 2 | or |Ψ ӿ Ψ| may be called the probability function. Since the electron must be Same where in space, the integration of |Ψ 2 | or |Ψ ӿ Ψ| Must be unity, so that Ψ ӿ (x) Ψ (x) dr Such wave function are said to be normalised.

6. Properties Of Ψ Acceptable wave function are those which satisfy the following conditions.

7. Postulate 2 “To every observable in classical mechanics there corresponds a linear operator in Quantum mechanics.” According to Postulate 2 , all quantum-mechanical operators are linear. There is an important property of linear operators that we have not discussed yet. Consider an eigenvalue problem with a two-fold degeneracy; that is, consider the two equations . ẬΨ 1 = aΨ 1 and ẬΨ 2 = aΨ 2 Both Ψ 1 and Ψ 2 have the same eigenvalue a.

8. If this is the case, then, any linear combination of Ψ 1 and Ψ 2 , say C 1 Ψ1 = C 2 Ψ 2, is also an eigenfunction of Ậ with the eigenvalue a. The proof relies on the linear property of Ậ . So, Ậ (C 1 Ψ1 = C 2 Ψ 2 ) = C 1 Ậ Ψ1+ C 2 Ậ Ψ 2 EXAMPLE

9. Example -- helps show that this result is directly due to the linear property of quantum-mechanical operators. Although we have considered only a two-fold degeneracy, the general result is obvious.

10. Postulate 3 “ The only possible values that can result from measurements of the physically observable property A are the eigenvalues a in the equation, ẬΨ = aΨ where Ậ is the operator corresponding to the property A . The eigenfunctions Ψ are required to be well behaved.” Our main concern is with the energy levels of atoms and molecules. These are given by the eigenvalues of the energy operator, the Hamiltonian Ĥ , The eigenvalue equation for Ĥ , Ĥ Ψ = EΨ , is the time-independent Schrödinger equation. However, finding the possible values of any property involves solving an eigenvalue equation.

11. We have, ẬΨ = aΨ . . . . . . . . . . . 1 Where Ậ and Ψ may be complex but a must be real. Here, we multiply equation 1 from the left by Ψ ӿ and integrate to obtain, ʃ Ψ ӿ ẬΨ dx = a ʃ Ψ ӿ Ψ dx = a . . . . . . . . . 2 Now, take the complex conjugate of equation 1 , Ậ ӿ Ψ ӿ = a ӿ Ψ ӿ =a Ψ ӿ . . . . . . . . . . .3 Where the equality a ӿ = a recognizes that a is real multiply equation 3 from the Left by Ψ and integrate, ʃ Ψ Ậ ӿ Ψ ӿ dx = a ӿ ʃ Ψ Ψ ӿ dx = a . .. . . . . .. .4

12. Equaling the left sides of equation 2 and 4 gives ʃ Ψ ӿ ẬΨ dx = ʃ ΨẬ ӿ Ψ ӿ dx . . . .. . . . . . . . . .5 The operater Ậ must satisfy equation 5 to assure that its eigenvalues are Real an operater that satisfies equation 5 for any well behaved function is called a Hermitian operater.

13. Postulate 4 “ If a System is in a state described by a normalized wave function Ψ, then the average value of the observable corresponding to Ậ is given by , ˂ a˃ = ʃ Ψ*ẬΨ dτ . . . . . . . 1 According to Postulate 4 , if we were to measure the energy of each member of a collection of similarly prepared systems, each described by Ψ , then the average of the observed values is given by Equation 1 with Ψ = Ĥ . Suppose that Ψ just happens to be an eigenfunction of Ậ ; that is, suppose that Ψ = Ψn Where Ậ Ψn = a n Ψn

14. Then ˂ a˃ = ʃ Ψ*Ậ Ψn dτ = ʃ Ψ* n a n Ψn dτ = a n ʃ Ψ* n Ψndτ = a n . .. . . . .2 Further more, if Ậ Ψn = a n Ψn then Ậ 2 Ψn = Ậ(Ậ Ψn)= Ậ(a n Ψn)= a n 2 Ψn . . . 3 And so ˂ a 2 ˃ = ʃ Ψ* n Ậ 2 Ψn dτ . .. . . . . .4 From Equations 3 and 4 , we see that the variance of the measurement gives, σ 2 n = ˂ a 2 ˃ - ˂ a˃ 2 = a n 2 - a n 2 = 0 . . . . . 5

15. Thus, Postulate 3 says that the only value that we measure is the value a n . Often, however, the system is not in a state described by an eigenfunction, and one measures distribution of values whose average is given by Postulate 4.