Chapter6-2
6.2 Quantum Mechanics
Quantum mechanics is based on the dual nature of matter. Experiments have verified that particles such as electrons show typical wavelike properties, including interference. Particles of small mass can be described by the wave function, , which represents the displacement of a wave as it fluctuates with time and position, = (q, t). The wave function can be expressed in the form
= Aexp(6.2-1)
where A is the amplitude of the wave, is the wavelength, and is the wave frequency. The wave function can be expressed in terms of energy E using Einstein’s formula
E = h = mc2 =
where h is Planck’s constant (h = 6.6310-34 Js), c is the velocity of light, and E is the particle or wave energy. The relations between frequency , wavelength , energy, and momentum p can be obtained from Einstein’s equation
=
= = =
The wavelength of particle predicted by = = agreed well with experimental data where v is the particle velocity. The wavelength of electron (m = = 9.10910-31 kg) travels at 0.05 speed of light is
== 4.8510-11 m
Substituting the frequency and wavelength in terms of energy and momentum into Eq. (6.2-1) yields
= Aexp = Aexpexp = AQ
where Q = exp, and = exp
Q and are called the spatial and temporal parts of the wave function, respectively. The wave function can be used to determine the system mechanical properties such as momentum p and energy E. These properties are obtained from differential operation on . From the definition of the wave function
= AQ, Q = exp, and = exp
Operation on the wave function yields the product of the mechanical variables (p or E) and the wave function as follows
= A = A exp =
= AQ = AQ exp
= E =
When an operator operates on a function to produce a constant time the original function, the constant is called an eigenvalue. For example, the operator operates on the wave function to produce a constant time the wave function. The constant is an eigenvalue. Similarly is an eigenvalue obtained from the operator operating on the wave function . The momentum and energy operators are then defined
=
=
The Hamiltonian operator is formulated by replacing the momentum in the Hamiltonian H(p,q)with the momentum operator.
H(p,q) = + V(q)
= + V = + V = 2 + V
For conservative system the Hamiltonian H is the energy E of the system, therefore the Hamiltonian operator is just the energy operator .
= =
When the Hamiltonian operator is used on the wave function, the resultant equation is called the Schrodinger equation that is the quantum mechanical equivalent of the equations of motion in classical mechanics. Schrodinger equation can be used to explain many physical phenomena at the atomic level.
= 2 + V
Let = , and = m, Schrodinger equation takes the following form
i = 2 + V(6.2-1)
The wave function only reflects the spatial probability of finding a particle. It does not reflect an actual mechanical location of a selected particle. The probability of finding a particle in an interval [a, b] is given by where (x) is called the probability density function. The function (x) is related to the wave function by the following expression
(x) = (x) (x) =
(x) is the complex conjugate of (x) as follow
(x) = Aexp(x) = Aexp
The probability of finding the particle somewhere in the entire x coordinate must be equal to 1. Hence
= = 1
The Schrodinger equation (6.2-1) can be solved if the potential V and the initial condition
(x, y, z, 0) = f(x, y, z)
are given. We assume that (x, y, z, t) can be separated into u(x, y, z), a function of spatial coordinates only, and T(t), a function of t alone.
(x, y, z, 0) = u(x, y, z) T(t)
In terms of the new functions, Eq. (6.2-1) becomes
iu = T2u + V uT
Dividing the equation by uT yields
i = 2u + V = E
where E is the separation constant. The differential equation on u is called the time-independent Schrodinger equation.
2u + Vu = Eu
Several simple solution of the Schrodinger equation can be used to explain the energy storage in atoms and molecules.
Example 6.2-1
Solve the Schrodinger equation for a small particle confined to move freely inside an interval [0, L] with potential
V(x) =
The initial wave function is known: (x, 0) = f(x)
Solution
The one-dimensional Schrodinger equation with V = 0 is
i = = E
The time dependent function is
i= E T =Aexp
The time independent function is
= Eu = 2u, where 2 =
For E > 0, the time independent wave function has the form
u(x) = acos(x) + bsin(x)
The constant a and b are determined from the boundary conditions that the probability of finding the particle must approach zero at x = 0 and x = L since a particle with finite energy cannot exist where V .
u(0) = 0 = a
u(L) = 0 = bsin(L) L = n2 = =
The separation constants Enare called the eigenvalues according to the definition
En =
The solutions un(x) are the eigenfunctions
un(x) = bn sin
The one-dimensional wave function is then
(x, t) = bn sin exp
The constants bn can be obtained from the initial condition
(x, 0) = f(x) = bn sin
bn =
Let consider the particular solutions of the one-dimensional time independent wave equation
u(x) = b sin
The constant b in this case can be determined from the requirement that the probability of finding the particle within the interval [0, L] must be equal to 1.
= 1 b2 = 1
Using the identity sin2x = , the above equation becomes
b2 = 1
= L = 1 b =
The full wave function solutions are therefore
u(x) = sin,n = 1, 2, 3, ...
For the 3-dimensional case, the time-independent equation becomes
= Eu
Using separation of variable and the requirement that
= 1
The 3-D solutions are
u(x, y, z) = sin sin sin
The eigenvalues for the solutions are
En = ( + + )
For a given Enthere are a number of different combinations of nx, ny, and nz. The wave function for each combination will be different but the total energy of the system will be the same. Hence the system may occupy a given energy level En in a number of different ways.
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