Why do lines become weaker in intensity as wavelength decreases? In 1885 the first series was observed by Swedish school teacher Johann Jakob Balmer in the visible region of the hydrogen spectrum. This series is called the Balmer series. In Balmer series as the wavelength decreases the lines appear closer together and are weaker in intensity. I understand why the lines appear closer together (because as the energy levels increase the gap between successive energy levels decreases). But why as wavelength decreases, the intensity of the lines are weaker? Can someone please explain?
 A: The intensity of spectral lines decreases as you move up the energy scale (or frequency scale) because there are fewer and fewer excited hydrogen atoms at high energies that decay directly to the ground state. There are other details as well, but this is roughly the main reason.
Quantum mechanically, the transition rate from the excited state to the ground state (which is responsible for the spectral lines) is dependent -- among other things -- on the square of the matrix element between the two states. Mathematically, if $|\Psi_0\rangle$ is the wavefunction of the ground state of hydrogen and $|\Psi_j\rangle$ is the wavefunction of the excited state $j>1$, then the matrix element between the two states is the inner product $\langle \Psi_j| \Psi_0\rangle$. Qualitatively, the matrix element represents how much overlap there is between the two states.
In other words, if the hydrogen atom is in the state $j$, then the probability with which it decays to the ground state is roughly proportional to the square of this overlap: $\mathcal{M} = |\langle \Psi_j| \Psi_0\rangle|^2$. As $j$ increases, this overlap decreases, leading to lower values of the transition rate. A lower transition rate means fewer atoms decaying directly to the ground state on average, which in turn leads to a lower intensity of that particular spectral line.
