Abstract
The properties of the time-independent Schrödinger equation are introduced step by step, starting from a short discussion about its boundary conditions. Considering that the equation is seldom amenable to analytical solutions, two simple cases are examined first: that of a free particle and that of a particle in a box. The determination of the lower energy bound follows, introducing more general issues that build up the mathematical frame of the theory: norm of a function, scalar product of functions, Hermitean operators, eigenfunctions and eigenvalues of operators, orthogonal functions, and completeness of a set of functions. The chapter is concluded with the important examples of the Hamiltonian operator and momentum operator. The complements provide examples of Hermitean operators, a collection of operators’ definitions and properties, examples of commuting operators, and a further discussion about the free-particle case.
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Notes
- 1.
It may happen that the domain is infinite in some direction and finite in the others. For instance, one may consider the case where w vanishes identically for \(x\ge 0\) and differs from zero for \(x<0\). Such situations are easily found to be a combination of cases A and B illustrated here.
- 2.
The one-to-one correspondence does not occur in general. Examples of the Schrödinger equation are easily given (Sect. 9.6) where to each eigenvalue there corresponds more than one—even infinite— eigenfunctions.
- 3.
Such a minimum is set to zero in the example of Sect. 8.2.1.
- 4.
The two terms \(\langle g \vert\) and \(\vert f \rangle\) of the scalar product \(\langle g \vert f \rangle\) are called bra vector and ket vector, respectively.
- 5.
In this context the term operator has the following meaning: if an operation brings each function f of a given function space into correspondence with one and only one function s of the same space, one says that this is obtained through the action of a given operator \({\cal A}\) onto f and writes \(s = {\cal A}f\). A linear operator is such that \({\cal A} \, (c_1 \, f_1 + c_2 \, f_2) = c_1 \, {\cal A} f_1 + c_2 \, {\cal A} f_2\) for any pair of functions f 1, f 2 and of complex constants c 1, c 2 ([78], Chap. II.11).
- 6.
The adjoint operator is the counterpart of the conjugate-transpose matrix in vector algebra.
- 7.
The completeness of a set of eigenfunctions must be proven on a case-by-case basis.
- 8.
The relation (8.44) is given here with reference to the specific example of the free particle’s eigenfunctions. For other cases of continuous spectrum the relation \(\langle w_\alpha \vert w_\beta \rangle = \delta (\alpha - \beta )\) is proven on a case-by-case basis.
- 9.
Consider for instance the calculation of the expectation value of the momentum of a free particle based on (10.18). If the minus sign were omitted in (8.48), the direction of momentum would be opposite to that of the propagation of the wave front associated to it.
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Problems
Problems
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8.1
The one-dimensional, time-independent Schrödinger equation is a homogeneous equation of the form
$$w^{\prime\prime} + q \, w = 0,\qquad q = q (x),$$(8.75)
where primes indicate derivatives. In turn, the most general, linear equation of the second order with a non-vanishing coefficient of the highest derivative is
Assume that a is differentiable. Show that if the solution of (8.75) is known, then the solution of (8.76) is obtained from the former by simple integrations.
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Rudan, M. (2015). Time-Independent Schrödinger Equation. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_8
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_8
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