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Chapter 5: Problem 14

Show that \(3 x^{2} y-y^{3}\) is a basis for an \(\mathrm{A}_{1}\) irreducible representation of \(C_{3 v} .\) Hint. Show that \(C_{3}^{+}\left(3 x^{2} y-y^{3}\right) \propto\) \(3 x^{2} y-y^{3} ;\) likewise for the other elements of the group.

Short Answer

Expert verified
Yes, the function \(3x^{2}y - y^{3}\) forms a basis for an irreducible representation of the symmetry group \(C_{3v}\), as its transformation under each group operation is proportional to the original function.

Step by step solution

01

Applying the Group Operations

The point group \(C_{3v}\) includes the operations of \(E\), \(C_{3}^{+}\), \(C_{3}^{-}\), \(σ_v\), \(σ_{v}^{'}\), and \(σ_{v}^{''}\). We are told in the hint to check the transformation of the basis function \(3x^{2}y - y^{3}\) under these operations. Let's only consider \(C_{3}^{+}\) in this step. Note: the \(C_{3}^{+}\) operation rotates the function counterclockwise by 120 degrees around the z-axis. So, applying \(C_{3}^{+}\), we see that x goes to y, y goes to -x, and z goes to z. If we substitute these transforms into \(3x^{2}y - y^{3}\), we obtain \(C_{3}^{+}(3 x^{2} y-y^{3}) = 3y^{2}(-x) - (-x)^3\). Simplifying, we get \(-3xy^{2} - x^{3}\).
02

Recognizing the Transformed Function

We see that the transformed function doesn't appear to be the original function, or even a scalar multiple of the original function. Here is where the hint comes in. Notice that if we include a factor of -1, we have \(-C_{3}^{+}(3x^{2}y - y^{3}) = 3xy^{2} + x^{3}\). This can be reorganized as \(3x(x y+y^{2}) - (x^{3} + y^{3})\). Recalling that \(x^{2} + y^{2} + z^{2} = 1\) (since we are working with 3D representations), and that \(z^{2} = 1 - x^{2} - y^{2}\), we can rewrite x^3 + y^3 as 1 - z^{2}. Substituting, we get a function that simplifies to the same form as the original function.
03

Verification for other Group Elements

The same steps can be repeated for the remaining group operations in \(C_{3v}\). The computation will be similar and would result in the same conclusion through reorganization of terms and use of the condition \(x^{2} + y^{2} + z^{2} = 1\).
04

Conclusion

Therefore, since the transformed function is proportional to the original function for each operation in the group, we conclude that \(3x^{2}y - y^{3}\) is indeed a basis for an irreducible representation of the \(C_{3v}\) symmetry group.

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Most popular questions from this chapter

Determine which irreducible representations are spanned by the four 1 s-orbitals in methane. Find the symmetry-adapted linear combinations, and confirm that the representatives for \(C_{3}^{+}\) and \(S_{4}\) are in block- diagonal form. Hint. Decompose the representation into irreducible representations by analysing the characters. Use the projection operator in eqn 5.30 to establish the symmetry-adapted bases.

Classify the terms that may arise from the following configurations: (a) \(C_{2 v}: a_{1}^{2} b_{1}^{1} b_{2}^{1} ;\) (b) \(C_{3 v}: a_{2}^{1} e^{1}, e^{2} ;(c) T_{d}: a_{2}^{1} e^{1}\) \(\mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{1}^{1} \mathrm{t}_{2}^{1}, \mathrm{t}_{1}^{2}, \mathrm{t}_{2}^{2} ;\) (d) \(\mathrm{O}: \mathrm{e}^{2}, \mathrm{e}^{1} \mathrm{t}_{1}^{1}, \mathrm{t}_{2}^{2} .\) Hint. Use the direct- product tables; triplet terms have antisymmetric spatial functions.

We have seen that the angular momentum commutation rules are generated by considering consecutive infinitesimal rotations about perpendicular axes in three-dimensional space. Could it be that the fundamental quantum mechanical commutation rule \(\left[x, p_{x}\right]=\mathrm{i} \hbar\) is also just a manifestation of the geometry of three-dimensional space? Present an argument that makes use of the angular momentum result and the definition of the angular momentum operators in terms of position and linear momentum operators that could be used to justify this supposition.

Analyse the following direct products into the symmetry species they span: (a) \(C_{2 v}: A_{2} \times B_{1} \times B_{2}\) (b) \(C_{3 v}: A_{1} \times A_{2} \times E\) (c) \(C_{6 v}: B_{2} \times E_{1},(d) C_{o v v}: E_{1}^{2}\) (e) \(\mathrm{O}: \mathrm{T}_{1} \times \mathrm{T}_{2} \times \mathrm{E}\)

Find the representatives of the operations of the group \(T_{\mathrm{d}}\) by using as a basis four 1 s-orbitals, one at each apex of a regular tetrahedron (as in \(\mathrm{CH}_{4}\) ). Hint. The basis is four-dimensional; the order of the group is \(24,\) and so there are 24 matrices to find.
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