HW 10: Rigid Rotor, Spherical Harmonics and Angular Momentum
Question-1 ¶ Using micrwave spectrum for structural information.
The microwave spectrum of 39 K 127 I ^{39}K ^{127}I 39 K 127 I
Question-2 ¶ Figuring out the rotational constant.
The equilibrium internuclear distance of H 127 I H^{127}I H 127 I B ~ \tilde{B} B ~
Question-3 ¶ Rotation-vibration transitions
Use the data in the following table for H 127 I H^{127}I H 127 I B 0 ~ \tilde{B_0} B 0 ~ B 1 ~ \tilde{B_1} B 1 ~ B e ~ \tilde{B_e} B e ~ α e ~ \tilde{\alpha_e} α e ~
Line [ B r a n c h ( t r a n i s i o n ) ] [Branch(tranision)] [ B r an c h ( t r ani s i o n )] Wave number [ c m − 1 ] [cm^{-1}] [ c m − 1 ] R ( 0 ) R(0) R ( 0 ) 2242.6 R ( 1 ) R(1) R ( 1 ) 2254.8 P ( 1 ) P(1) P ( 1 ) 2217.1 P ( 2 ) P(2) P ( 2 ) 2203.8
Question-4 ¶ Quantum numbers: ( l , m ) ({\bf l},m) ( l , m ) First few spherical harmonics: Y l m ( θ , ϕ ) Y^m_l(\theta,\phi) Y l m ( θ , ϕ ) ( 0 , 0 ) ({\bf 0},0) ( 0 , 0 ) 1 ( 4 π ) 1 / 2 \frac{1}{(4\pi)^{1/2}} ( 4 π ) 1/2 1 ( 1 , 0 ) ({\bf 1},0) ( 1 , 0 ) ( 3 4 π ) 1 / 2 c o s θ \Big(\frac{3}{4\pi}\Big)^{1/2}cos\theta ( 4 π 3 ) 1/2 cos θ ( 1 , 1 ) ({\bf 1},1) ( 1 , 1 ) ( 3 8 π ) 1 / 2 s i n θ e i ϕ \Big(\frac{3}{8\pi}\Big)^{1/2} sin\theta e^{i\phi} ( 8 π 3 ) 1/2 s in θ e i ϕ ( 1 , − 1 ) ({\bf 1},-1) ( 1 , − 1 ) ( 3 8 π ) 1 / 2 s i n θ e − i ϕ \Big(\frac{3}{8\pi}\Big)^{1/2}sin\theta e^{-i\phi} ( 8 π 3 ) 1/2 s in θ e − i ϕ ( 2 , 0 ) ({\bf 2},0) ( 2 , 0 ) ( 5 16 π ) 1 / 2 ( 3 c o s 2 θ − 1 ) \Big(\frac{5}{16\pi}\Big)^{1/2}(3cos^2\theta-1) ( 16 π 5 ) 1/2 ( 3 co s 2 θ − 1 ) ( 2 , 1 ) ({\bf 2},1) ( 2 , 1 ) ( 15 8 π ) 1 / 2 s i n θ cos θ e i ϕ \Big(\frac{15}{8\pi}\Big)^{1/2}sin\theta \cos\theta e^{i\phi} ( 8 π 15 ) 1/2 s in θ cos θ e i ϕ ( 2 , − 1 ) ({\bf 2},-1) ( 2 , − 1 ) ( 15 8 π ) 1 / 2 s i n θ cos θ e − i ϕ \Big(\frac{15}{8\pi}\Big)^{1/2}sin\theta \cos\theta e^{-i\phi} ( 8 π 15 ) 1/2 s in θ cos θ e − i ϕ ( 2 , 2 ) ({\bf 2},2) ( 2 , 2 ) ( 15 32 π ) 1 / 2 s i n 2 θ e 2 i ϕ \Big(\frac{15}{32\pi}\Big)^{1/2}sin^2\theta e^{2i\phi} ( 32 π 15 ) 1/2 s i n 2 θ e 2 i ϕ ( 2 , − 2 ) ({\bf 2},-2) ( 2 , − 2 ) ( 15 32 π ) 1 / 2 s i n 2 θ e − 2 i ϕ \Big(\frac{15}{32\pi}\Big)^{1/2}sin^2\theta e^{-2i\phi} ( 32 π 15 ) 1/2 s i n 2 θ e − 2 i ϕ
Give one example of normalization and orthogonality of spherical harmonics. [Use any pair from the table in the book which has not already been done in the class.]
How many nodal lines along θ \theta θ ϕ \phi ϕ ( 2 , 2 ) (2, 2) ( 2 , 2 )
Demonstrate the Unso ¨ \ddot{o} o ¨
∑ m = − l m = l ∣ Y l m ( θ , ϕ ) ∣ 2 = c o n s t a n t \sum^{m=l}_{m=-l} \mid Y^{m}_{l}(\theta,\phi) \mid^2=constant m = − l ∑ m = l ∣ Y l m ( θ , ϕ ) ∣ 2 = co n s t an t Question-5 :books: ¶ Angular momentum and spatial quantization.
What is the significance of angular momentum in physics/chemistry?
What is the meaniing of the commuitation relation [ L x ^ , L y ^ ] = i ℏ L z [\hat{L_x}, \hat{L_y}]=i\hbar L_z [ L x ^ , L y ^ ] = i ℏ L z
Why is angular momentum in Qm often portrayed as a vector rotating around z axis with different discrete values of an angle with respect to z.
Using three spherical harmonics Y 1 0 , Y 1 + 1 , Y 1 − 1 , Y^0_1,Y^{+1}_1,Y^{-1}_1, Y 1 0 , Y 1 + 1 , Y 1 − 1 , L z L_z L z
Given the eigenvalue of L ^ \hat{L} L ^ ℏ 12 \hbar\sqrt{12} ℏ 12 L z ^ \hat{L_z} L z ^