A Rydberg atom (discussed in more detail in Chapter 8) is a single-electron atom with a...

A Rydberg atom (discussed in more detail in Chapter 8) is a single-electron atom with a large quantum number n. Rydberg states are close together in energy (see Figure 4.15), so transitions between adjacent Rydberg states produce long-wavelength photons. Consider a transition from a state n + 1 to a state n in hydrogen. (a) Starting with Equation (4.30), use the binomial expansion to show that this transition produces a photon with wavelength approximately n³/2R. (b) Obtain the same result as in part (a), this time starting with Equation (4.25) and computing dE/dn. The result, dE/dn, can then be approximated by ΔE/ Δn, with Δn = 1 for this transition and ΔE =hc/ λ for the emitted photon. (c) Using the approximate expression you derived in (a) and (b), compute the wavelength for a transition from n = 101 to n =100 in hydrogen. (Use R_∞and ignore the reduced-mass correction.) Compare your answer with the exact wavelength for this transition, computed using Equation (4.30).