the quantum state or energy is small. The applied external field causes the change in
Hamiltonian of a quantum system and results in the change in that quantum state
usually accompanied by the energy change of the system. We describe this process
as a following eigenvalue equation:
H j Ψ i i ¼ E i j Ψ i i,
ð5:1Þ
where H represents the total Hamiltonian; E i is an energy eigenvalue and Ψ i is an
eigenstate corresponding to E i . Equation (5.1) implies that there should be a series of
eigenvalues and corresponding eigenvectors belonging to those eigenvalues. Strictly
speaking, however, we can get such combinations of the eigenstate and eigenvalue
only by solving (5.1) analytically. This is some kind of circular argument. Yet, at
present we do not have to go into detail about that issue. Instead we advance to
discussion of the approximation methods from a practical point of view.
We assume that H is divided into two terms such that
H ¼ H 0 þ λV,
ð5:2Þ
where H 0 is the Hamiltonian of the unperturbed system without the external field;
V is an additional Hamiltonian due to the applied external field, or interaction
between the physical system and the external field. The second term includes a
parameter λ that can be modulated according to the change in the strength of the
applied field. The parameter λ should be real so that H of (5.2) can be Hermitian. The
parameter λ can be the applied field itself or can merely be a dimensionless parameter
that can be set at λ ¼ 1 after finishing the calculation. We will come back to this point
later in Examples.
In (5.2) we assume that the following equation holds:
H 0 j ii ¼ E
0
ð Þ
i j ii,
ð5:3Þ
where jii is the i-th quantum state. We designate j0i as the ground state. We assume
the excited states ji i (i ¼ 1, 2, Á Á Á) to be numbered in order of increasing energies.
These functions (or vectors) jii (including j0i) can be a quantum state that appeared
as various eigenfunctions in the previous chapters. Of these, two or more functions
may have the same eigenenergy (i.e., degenerate). If we are to deal with such
degenerate states, those states jii have to be distinguished by, e.g., ji d i
(d ¼ 1, 2, Á Á Á, s), where s denotes the degeneracy (or degree of degeneracy). For
simplicity, however, in this chapter we are going to examine only the change in
energy and physical states with respect to nondegenerate states. We disregard the
issue on notation of the degenerate states accordingly. We pay attention to each
individual case later, when necessary.
Regarding a one-dimensional physical system, e.g., a particle confined within
potential well and quantum-mechanical harmonic oscillator, for example, all the
quantum states are nondegenerate as we have already seen in Chaps. 1 and 2. As a
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5 Approximation Methods of Quantum Mechanics
Hamiltonian of a quantum system and results in the change in that quantum state
usually accompanied by the energy change of the system. We describe this process
as a following eigenvalue equation:
H j Ψ i i ¼ E i j Ψ i i,
ð5:1Þ
where H represents the total Hamiltonian; E i is an energy eigenvalue and Ψ i is an
eigenstate corresponding to E i . Equation (5.1) implies that there should be a series of
eigenvalues and corresponding eigenvectors belonging to those eigenvalues. Strictly
speaking, however, we can get such combinations of the eigenstate and eigenvalue
only by solving (5.1) analytically. This is some kind of circular argument. Yet, at
present we do not have to go into detail about that issue. Instead we advance to
discussion of the approximation methods from a practical point of view.
We assume that H is divided into two terms such that
H ¼ H 0 þ λV,
ð5:2Þ
where H 0 is the Hamiltonian of the unperturbed system without the external field;
V is an additional Hamiltonian due to the applied external field, or interaction
between the physical system and the external field. The second term includes a
parameter λ that can be modulated according to the change in the strength of the
applied field. The parameter λ should be real so that H of (5.2) can be Hermitian. The
parameter λ can be the applied field itself or can merely be a dimensionless parameter
that can be set at λ ¼ 1 after finishing the calculation. We will come back to this point
later in Examples.
In (5.2) we assume that the following equation holds:
H 0 j ii ¼ E
0
ð Þ
i j ii,
ð5:3Þ
where jii is the i-th quantum state. We designate j0i as the ground state. We assume
the excited states ji i (i ¼ 1, 2, Á Á Á) to be numbered in order of increasing energies.
These functions (or vectors) jii (including j0i) can be a quantum state that appeared
as various eigenfunctions in the previous chapters. Of these, two or more functions
may have the same eigenenergy (i.e., degenerate). If we are to deal with such
degenerate states, those states jii have to be distinguished by, e.g., ji d i
(d ¼ 1, 2, Á Á Á, s), where s denotes the degeneracy (or degree of degeneracy). For
simplicity, however, in this chapter we are going to examine only the change in
energy and physical states with respect to nondegenerate states. We disregard the
issue on notation of the degenerate states accordingly. We pay attention to each
individual case later, when necessary.
Regarding a one-dimensional physical system, e.g., a particle confined within
potential well and quantum-mechanical harmonic oscillator, for example, all the
quantum states are nondegenerate as we have already seen in Chaps. 1 and 2. As a
152
5 Approximation Methods of Quantum Mechanics