Chapter 5
Approximation Methods of Quantum
Mechanics
In Chaps. 1–3 we have focused on solving eigenvalue equations with respect to a
particle confined within a one-dimensional potential well or a harmonic oscillator
along with an electron of a hydrogen-like atom. In each example we obtained exact
analytical solutions with the quantum-mechanical states and corresponding eigenvalues (energy, angular momentum, etc.). In most cases of quantum-mechanical
problems, however, we are not able to get such analytical solutions or accurately
determine the corresponding eigenvalues. Under these circumstances, we need
appropriate approximation methods of those problems. Among those methods, the
perturbation method and variational method are widely used.
In terms of usefulness, we provide several examples concerning physical systems
that have already appeared in Chaps. 1–3. In this chapter, we examine how these
physical systems change their quantum states and corresponding energy eigenvalues
as a result of undergoing influence from the external field. We assume that the
change results from the application of external electric field. For simplicity, we focus
on the change in eigenenergy and corresponding eigenstate with respect to the
nondegenerate quantum state. As specific cases in these examples, we happen to
be able to get perturbed physical quantities accurately. Including such cases, for later
purposes we take in advance important concepts of a complete orthonormal system
(CONS) and projection operator.
5.1 Perturbation Method
In Chaps. 1–3, we considered a situation where no external field is exerted on a
physical system. In Chap. 4, on the other hand, we studied the optical transition that
takes place as a consequence of the interaction between the physical system and
electromagnetic wave. In this chapter, we wish to examine how the physical system
changes its quantum state, when an external electric field is applied to the system.
We usually assume that the external field is weak and the corresponding change in
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_5
151
Approximation Methods of Quantum
Mechanics
In Chaps. 1–3 we have focused on solving eigenvalue equations with respect to a
particle confined within a one-dimensional potential well or a harmonic oscillator
along with an electron of a hydrogen-like atom. In each example we obtained exact
analytical solutions with the quantum-mechanical states and corresponding eigenvalues (energy, angular momentum, etc.). In most cases of quantum-mechanical
problems, however, we are not able to get such analytical solutions or accurately
determine the corresponding eigenvalues. Under these circumstances, we need
appropriate approximation methods of those problems. Among those methods, the
perturbation method and variational method are widely used.
In terms of usefulness, we provide several examples concerning physical systems
that have already appeared in Chaps. 1–3. In this chapter, we examine how these
physical systems change their quantum states and corresponding energy eigenvalues
as a result of undergoing influence from the external field. We assume that the
change results from the application of external electric field. For simplicity, we focus
on the change in eigenenergy and corresponding eigenstate with respect to the
nondegenerate quantum state. As specific cases in these examples, we happen to
be able to get perturbed physical quantities accurately. Including such cases, for later
purposes we take in advance important concepts of a complete orthonormal system
(CONS) and projection operator.
5.1 Perturbation Method
In Chaps. 1–3, we considered a situation where no external field is exerted on a
physical system. In Chap. 4, on the other hand, we studied the optical transition that
takes place as a consequence of the interaction between the physical system and
electromagnetic wave. In this chapter, we wish to examine how the physical system
changes its quantum state, when an external electric field is applied to the system.
We usually assume that the external field is weak and the corresponding change in
© Springer Nature Singapore Pte Ltd. 2020
S. Hotta, Mathematical Physical Chemistry,
https://doi.org/10.1007/978-981-15-2225-3_5
151