Eq. (3.6) already demonstrated that in a mixed state of Eigenvalues, not belonging to the same energy-Eigenvalue, the fraction of each state changes as a function of time. The consequences for this changing state shall be discussed in what follows. As an example we take a state of two Eigenfunctions with different energies:
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| \begin{equation*} \psi(t, \vec{r}) = c_1 f_1(\vec{r}) e^{i \omega_1 t} + c_2 f_2(\vec{r}) e^{i \omega_2 t} \end{equation*} | (3.14) |
For the probability density we find
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| \begin{equation*} \psi^*(t, \vec{r}) \psi(t, \vec{r}) = |c_1 f_1(\vec{r})|^2 + |c_2 f_2(\vec{r})|^2 + 2 c_1 c_2 f_1 f_2 \cos (\omega_1 - \omega_2) t \end{equation*} | (3.15) |
The probability density of the electron will thus change harmonically with the rotation frequency \( (\omega_1 - \omega_2)\). This will in generally lead to a periodic change in the charge distribution, respectively of the dipole moment of the atom. Since the charge density is defined by
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| \begin{equation*} e \psi^*(t, \vec{r}) \psi(t, \vec{r}) d^3r \qquad , \end{equation*} | (3.16) |
we find for the complete dipole moment
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| \begin{equation*} \langle \vec{P}\rangle = e \int \psi^*(t, \vec{r}) \vec{r} \psi(t, \vec{r}) d^3r\qquad . \label{dipolemom} \end{equation*} | (3.17) |
Defining the matrix elements
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| \begin{equation*} \vec{M}_{i j} = e \int f_i^*(\vec{r}) \vec{r} f_j(\vec{r}) d^3r \qquad , \label{m_elements} \end{equation*} | (3.18) |
Eq. (3.17) can be rewritten as
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| \begin{equation*} \langle \vec{P}\rangle = |c_1|^2 \vec{M}_{1 1} + |c_2|^2 \vec{M}_{2 2} + 2 c_1 c_ 2 \vec{M}_{1 2} \cos (\omega_2- \omega_1) t \qquad . \end{equation*} | (3.19) |
i.e. the dipole moment of the atom will change periodically. Consequently the atom serves as a sender for electromagnetic radiation with the frequency \( (\omega_2- \omega_1)\). It will loose power in the state with the higher energy. After a short time the atom will be in the ground state which is not a mixed state.
This is the quantum mechanical explanation for the formula
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| \begin{equation*} \Delta E = h \nu \qquad . \end{equation*} | (3.20) |
This explains too, why atoms can be stable, although they have a rotational momentum (in the classical description they would always radiate light and thus be destroyed). This classical explanation results from the wrong picture, that the electron is moving through the orbital, leading to a steady change in the dipole moment.
Each state which is not an Eigenstate of the Hamiltonian has a non infinite-lifetime, which is defined in our example by the matrix element in Eq. (3.18).
The calculation of these matrix elements is the main task of quantum mechanics.
Non-zero matrix elements (even if the corresponding states are not degenerate) define rules for allowed transitions; e.g. radiative transitions only occur when \(\Delta l = \pm 1\) holds.
In addition to the radiative transitions there exist a large number of different mechanisms for an excited state to loose energy. The corresponding matrix elements (cross sections) may differ extremely.
If the probability for an excited state to loose energy is very small, the state is called metastable.
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© J. Carstensen (Quantum Mech.)