Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 11
Quantum- Mechanical Operators Represent ClassicalMechanical Variables
Postulate 2
To every observable in classical mechanics there
corresponds an operator in quantum mechanics.
Table (1)
Classical-Mechanical Observables and their corresponding
quantum-mechanical operators
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Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 11
Postulate 3
In any measurement of the observable associated with the
operator Â, the only values that will ever be observed are the
eigenvalues a, which satisfy the eigenvalue equation
A  a
(25)
As a specific example, consider the measurement of the energy.
The operator corresponding to the energy is the Hamiltonian
operator, and its eigenvalue equation is
H  n  E n n
(26)
This is just the Schrodinger equation. The solution of this
equation gives the  n and E n . For the case of a particle in a box,
En 
n 2h 2
8ma 2
. Postulate 3 says that if we measure the energy of a
particle in a box, we shall find one of these energies and no
others.
Postulate 4
If a system in a state described by a normalized wave
function ψ, then the average value of the observable
corresponding to  is given by

a    A d 

(27)
According to postulate 4, if we were to measure the energy of
each member of a collection of similarly prepared system, each
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Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 11
described by ψ, then the average of the observed values is
given by equation 27 with Â=Ĥ.
The Time Dependence of wave Function is governed by the
Time – Dependent Schrodinger Equation
Postulate 5
The wave function or state function of a system evolves in
time according to the time- dependent Schrodinger equation
H  (x , t )  i

t
(28)
We can write ψ(x, t) =ψ(x) f (t)
Also we take before Schrödinger equation
Ĥ ψ (x) =E ψ (x)
From these Eqs. We can write f (t) = e –i Et/ħ
 (x , t )   (x )e iEt /
(29)
 n (x , t )   n (x )e  iEn /
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(30)
Dr.Eman Zakaria Hegazy Quantum Mechanics and Statistical Thermodynamics Lecture 11
If the system happens to be in one of the eigenstates given by
Eq. 30,then
 n (x , t ) n (x , t )dx   n (x ) n (x )dx
(31)
Thus, the probability density and the averages calculated from
eq.30 are independent of time, and the ψ n(x) are called
stationary –state wave functions. The Bohr model of the
hydrogen atom is a simple illustration of this idea.
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