Solution of Certain Problems in Quantum Mechanics
By A. Bolotin, A. Pozamantir and Raudeliunas, A.
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About this ebook
Starting with a brief introduction to the hypergeometric equations and their properties, a step-by-step method consisting of six distinct parts illustrates how to address typical problems in quantum physics in a simple and uniform fashion. This technique can also be applied to the solution of other problems, for which the Schrödinger equation can be reduced by some means to an equation of the hypergeometric type. Topics include the discrete spectrum eigenfunctions, linear harmonic oscillators, Kratzer molecular potential, and the rotational correction to the Morse formula. The text concludes with an Appendix that presents an original Fourier transform-based method for converting multicenter integrals to a single center.
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Solution of Certain Problems in Quantum Mechanics - A. Bolotin
manuscript.
Chapter 1
THE HYPERGEOMETRIC FUNCTION
Many basic problems of quantum mechanics, such as the motion of a particle in a centrally-symmetric field, the linear harmonic oscillator, solving the Schrödinger, Dirac and Klein-Gordon equations for a Coulomb potential, and the motion of a particle in a uniform electric and magnetic field, lead to the differential equation
where σ(x) is a polynomial of degree no higher than one.
A number of model problems in atomic, molecular, and nuclear physics, such as solving the Schrödinger equation with the Morse, Kratzer, Poschl-Teller potentials, etc., also result in equation (1.1). Later in the text we will show that the classical orthogonal polynomials (Jacoby, Lagguere, Hermite polynomials), spherical, cylindrical, and hypergeometric functions, often referred to as special functions, are all particular solutions of equation (1.1).
Throughout this book, we will assume that the variable x and the coefficients of the polynomials σ(x)can have either real or complex values.
Examples
1. The well-known Schrödinger equation for the linear harmonic oscillator has the form
We see that equation (1.2) is a special case of equation (1.1) with σ(x) = 1,
Indeed, after bringing (1.2) to file:///D:/_KURAL/E-Others/Courier/99/Silk/QC/OEBPS/img/12_3.jpga common denominator and multiplying it by −1, it becomes
2. The differential equation for the spherical harmonics resulting from the Schrödinger equation for a centrally-symmetric potential is
where x = cos θ. One can see that
3. It is known that the radial part of the Schrödinger equation for a particle in a Coulomb field is (assuming the elementary charge e = 1)
After bringing it to a common denominator and multiplying by −1, we have
It follows that
By making the substitution u = φ(x)y and choosing the function φ(x) in a certain way, one can simplify equation (1.1) to read
where τ(x) is a polynomial of degree no higher than one, and λ is a constant.
In particular, if we choose u = φ(x)y, then
and
After substituting these into (1.1), we arrive at
or, upon dividing this equation by φ(x),
For equation (1.6) to be no more complex than the initial equation (1.1), we require that the coefficient at y′ be of the form τ(x), where τ(x) is a polynomial of degree no higher than one:
Then
where π(x) is a polynomial of degree no higher than one. From (1.7) we find
Since
equation (1.6) becomes
Let us transform the coefficient of y in equation (1.9):
where
After substituting (1.10) into equation (1.9), we have
It follows from equations (is a polynomial of degree no higher than two. Therefore, (1.12) is an equation of the same type as (1.1). In other words, we have found a transformation class that does not alter the type of equation. Let us make use of the freedom of an arbitrary choice of the polynomial π(x) to maximally simplify the form of equation (1.12). To this end, we choose the coefficients of the polynomial π(x) from (1.12) is divisible (without remainder) by the polynomial σ(x), so