91Ó°ÊÓ

An electron is confined to a one-dimensional potential well of width \(6 \times 10^{-10} \mathrm{~m}\) which has infinitely high sides. Calculate: (i) the three lowest allowed values of the electron energy; (ii) the wavelength of the electromagnetic wave that would cause the electron to be excited from the lowest to the highest of these three levels; (iii) all possible wavelengths of the radiation emitted following the excitation in (ii).

Consider a particle of mass \(m\) subject to a one-dimensional potential \(V(x)\) that is given by $$ V=\infty, \quad x<0 ; \quad V=0, \quad 0 \leqslant x \leqslant a ; \quad V=V_{0}, \quad x>a $$ Show that bound \(\left(E

Show that if \(V(x)=V(-x)\), solutions to the time-dependent Schrödinger equation must have definite parity-that is, \(u(x)=\pm u(-x)\). Hint: Make the substitution \(y=-x\) and show first that \(u(x)=A u(-x)\) where \(A\) is a constant.

The hydrogen atom in a water molecule can vibrate in a direction along the \(\mathrm{O}-\mathrm{H}\) bond, and this motion can be excited by electromagnetic radiation of a wavelength about \(4 \times 10^{-6} \mathrm{~m}\), but not by radiation of a longer wavelength. Calculate the effective spring constant for this vibration and the zero-point energy of the oscillator. Given that every molecular degree of freedom has a thermal energy of about \(k_{B} T\), where \(k_{B}\) (Boltzmann's constant) \(\simeq 1.4 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}\) and \(T\) is the temperature, what is the most probable vibrational state in the case of a water molecule in steam at \(450 \mathrm{~K}\) ?