1000 Solved Problems in Modern Physics

144 3 Quantum Mechanics – II conditions to sketch the form ofψ(x) in the region aroundx =0for the cases (i) and (ii). 3.42 A ste ...

3.2 Problems 145 boundaries, write down general expressions for the wavefunctions in these regions and the form the time-indepen ...

146 3 Quantum Mechanics – II 3.2.4 Simple Harmonic Oscillator ......................... 3.51 Show that the wavefunctionψ 0 (x) = ...

3.2 Problems 147 reasoning briefly. (a) If the transitions are vibrational, estimate the spring con- stant (in dyne/cm) (b) If t ...

148 3 Quantum Mechanics – II 3.65 Show that (a) the electron density in the hydrogen atom is maximum atr=a 0 , wherea 0 is the B ...

3.2 Problems 149 3.2.6 Angular Momentum ................................ 3.77 Given thatL=r×p, show that [Lx,Ly]=iLz 3.78 The s ...

150 3 Quantum Mechanics – II 3.85 (a) Obtain the angular momentum matrices forj= 1 (b) Hence obtain the matrix forJ^2. 3.86 Two ...

3.2 Problems 151 3.91 The normalized 2peigen functions of hydrogen atom are 1 √ π 1 (2a 0 )^3 /^2 e−r/^2 a^0 r 2 a 0 sinθeiΦ, 1 ...

152 3 Quantum Mechanics – II 3.2.7 Approximate Methods ......................... 3.97 Consider hydrogen atom with proton of fini ...

3.2 Problems 153 3.102 A particle of mass m is trapped in a potential well which has the form, (V=^1 /^2 mω^2 x^2. Use the varia ...

154 3 Quantum Mechanics – II show thatσ(θ)=a^2 [ 1 −(ka) 2 3 +2(ka) (^2) cosθ+··· ] andσ= 4 πa^2 [1−(ka)^2 /3] 3.110 Find the el ...

3.2 Problems 155 3.115 Given the scattering amplitude f(θ)=(1/ 2 ik) ∑ (2l+1) [ e^2 iδl− 1 ] Pl(cosθ) Show that Im f(0)=kσt/ 4 π ...

156 3 Quantum Mechanics – II 3.123 Using the Born approximation, the amplitude of scattering by a spherically symmetric potentia ...

3.3 Solutions 157 3.3 Normalization condition is ∫∞ −∞ |ψ|^2 dx= 1 N^2 ∫∞ −∞ (x^2 +a^2 )−^2 dx= 1 Putx=atanθ;dx=sec^2 θdθ ( 2 N^ ...

158 3 Quantum Mechanics – II This is the continuity equation where the probability currentJ= 2 im^1 (φ∗∇φ− φ∇φ∗) And probability ...

3.3 Solutions 159 (b) The difference between the wave functions in the infinite and finite poten- tial wells is that in the form ...

160 3 Quantum Mechanics – II wherePlm(cosθ) are the associated Legendre functions Now, Plm(cosθ)=(1−cos^2 θ)m/^2 dmPl(cosθ)/dcos ...

3.3 Solutions 161 3.9 (ψ,Qψ)=(ψ,qψ)=q(ψ,ψ) (Qψ,ψ)=(qψ,ψ)=q∗(ψ,ψ) sinceQis hermitian, (ψ,Qψ)=(Qψ,ψ) and thatq=q∗ That is, the eig ...

162 3 Quantum Mechanics – II 3.12 First the wave function is normalized N^2 ∫∞ 0 ψ∗ψdx= 1 N^2 ∫∞ 0 (√ 2 e− x L ) 2 dx= 1 N= 1 / ...

3.3 Solutions 163 From the recurrence relation (6) a 1 =− a 0 2 Therefore,y(r)=a 0 r ( 1 −r 2 ) F(r)=a 0 e− r (^2) r ( 1 − r 2 ) ...