(a_)*(a_V)ax- J-oProblem 2.11 Show that the lowering operator cannot generate a state of infinitenorm (i.e., f la.-²dx < oo, if y itself is a normalized solution to the Schrödingerequation). What does this tell you in the case y = vo? Hint: Use integration byparts to show thaty*(a,a_) dx.=-00Then invoke the Schrödinger equation (Equation 2.46) to obtainla-yl² dx E - hw,-0-where E is the energy of the state y.**Problem 2.12(a) The raising and lowering operators generate new solutions to the Schrödingerequation, but these new solutions are not correctly normalized. Thus a Vnis proportional to yn+1, and a n is proportional to yn-1, but we'd like toknow the precise proportionality constants. Use integration by parts and theSchrödinger equation (Equations 2.43 and 2.46) to show thatroo| la+ Vl² dx = (n+ 1)hw,la- Vnl? dx = nhw,-00-00and hence (with i's to keep the wavefunctions real)a+ Vn = iv(n + 1)hw yn+1,[2.52]a_n = -ivnhw n-1.[2.53]Sec. 2.3: The Harmonic Oscillator37(b) Use Equation 2.52 to determine the normalization constant A, in Equation 2.50.(You'll have to normalize o "by hand".) Answer:1/4(-i)"An =[2.54]Vn!(hw)"*Problem 2.13 Using the methods and results of this section,(a) Normalize n (Equation 2.51) by direct integration. Check your answer againstthe general formula (Equation 2.54).(b) Find 2, but don't bother to normalize it.

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Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that < 0o, if y itself is a normalized solution to the Schrödinger | (a-v)*(a_v) dx = v*(a,a_v)dx. 00 Then invoke the Schrödinger equation (Equation 2.46) to obtain 1. la-v* dx = E -ħw, 2 where E is the energy of the state y.

Problem 2.11 Show that the lowering operator cannot generate a state of infinite norm (i.e., f la.-²dx < equation). What does this tell you in the case y = vo? Hint: Use integration by parts to show that < 0o, if y itself is a normalized solution to the Schrödinger | (a-v)(a_v) dx = v(a,a_v)dx. 00 Then invoke the Schrödinger equation (Equation 2.46) to obtain 1. la-v* dx = E -ħw, 2 where E is the energy of the state y.