Abstract
Symmetry means simply that there are different ‘perspectives’ from which a physical system looks the same. In other words, the system is invariant under a certain symmetry operation such as rotation or reflection, and its mathematical description does not change as a result of the transformation.
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Notes
- 1.
In general, there are two viewpoints in considering symmetry transformations: the passive viewpoint (the system remains unchanged, but the axes are changed accordingly), and the active viewpoint (the axes remain unchanged, but the system is transformed—which is of course not meant to be a dynamic rotation). Which point of view one prefers is a matter of taste. The best-known example is perhaps the passive and active rotation in two dimensions, which we already mentioned in Chap. 2, Vol. 1.
- 2.
P.W. Anderson, Nobel Laureate 1972: “It is only a slight exaggeration to say that physics is the study of symmetries.”
- 3.
In special problems, there may of course be additional symmetries (e.g. the Lenz vector in the case of the hydrogen atom).
- 4.
Galilean and not Lorentz transformations, because we are considering here nonrelativistic quantum mechanics.
- 5.
Even Wolfgang Pauli took the validity of the symmetries for granted and consequently declared it as a priori absurd to search for parity-violating processes: “I cannot believe that God is a weak left-hander.” He had to revise his views, as is known, after Madame Wu et al., demonstrated parity violation in the beta decay of cobalt-60 atoms in 1956.
- 6.
Number of parameters: \(3 +1 +3 +3 = 10\).
- 7.
For example, we have seen in an exercise for Chap. 9, Vol. 1 that the angular momentum is conserved in spherically symmetric problems.
- 8.
The full version can be found in Appendix I, Vol. 1.
- 9.
This relationship is suggested, inter alia, by the fact that a function \(f(x) \) with the property \(f(x+y)=f(x)f(y)\) is given by a (generalized) exponential function \(a^{x}\).
- 10.
The sign and any other multiplicative constants cannot be determined at this point (\(e^{i\alpha T_{S}}\), \(e^{-i\alpha T_{S}/\hbar }\), etc.).
- 11.
Why is one not content with \(\left[ U_{S},H\right] =0\)? The answer is because \(U\) is unitary, but \(T\) is Hermitian and therefore is possibly a measurable variable.
- 12.
We repeat the remark that a unitary transformation corresponds to a change of basis in \(\mathcal {H}\).
- 13.
Using as example the matrix representation of \(e^{-i\frac{\gamma {\mathbf {j}}\hat{\mathbf {a}}}{\hbar }}\) for orbital angular momentum \(\mathbf {l}=1\); see Appendix W.2, Vol. 2.
- 14.
More on vector operators is found in Appendix L, Vol 2.
- 15.
If \(\mathcal {T}^{2}=1\), then the transformation clearly cannot come from a continuous group. For instance, we have for a rotation \(\left( \mathcal {T}_{\alpha }\right) ^{2}=\mathcal {T}_{2\alpha }\ne 1\) for arbitrary angles \(\alpha \).
- 16.
Because of e.g. \(\mathcal {P}\mathbf {r}f\left( \mathbf {r}\right) =-\mathbf {r} \mathcal {P}f(\mathbf {r})\); from this, it follows that \(\mathcal {P}\mathbf {r} =-\mathbf {r}\mathcal {P}\), and thus \(\mathcal {P}\mathbf {r}\mathcal {P}=- \mathbf {r} \).
- 17.
Of course, quite generally every state \(\psi \left( \mathbf {r}\right) \) may be divided into an even and an odd part: \(\psi \left( \mathbf {r}\right) =\psi _{+}\left( \mathbf {r}\right) +\psi _{-}\left( \mathbf {r}\right) \), with \(\psi _{\pm }\left( \mathbf {r}\right) =\frac{\psi \left( \mathbf {r}\right) \pm \psi \left( -\mathbf {r}\right) }{2}\).
- 18.
Because of \(\mathbf {l}=\mathbf {r}\times \mathbf {p}\rightarrow \mathbf {l}=\left( -\mathbf {r}\right) \times \left( -\mathbf {p}\right) =\mathbf {r}\times \mathbf {p}\). Generally speaking, the product of two polar vectors is a pseudovector.
- 19.
When forces are conservative, the orbits are time-reversal invariant. Magnetic fields are not conservative.
- 20.
We repeat the remark that an anti-unitary operator \(B\) is anti-linear, i.e. \(B\!B^{\dag }=B^{\dag }\!B=1\) and \(B\alpha \left| \varphi \right\rangle =\alpha ^{*}B\left| \varphi \right\rangle \).
- 21.
The considerations for time-dependent Hamiltonians are more complex and require new concepts (time-ordering operator, etc.); but time reversal is an anti-unitary operator in this case, also.
- 22.
In general, every anti-unitary operator \(A\) may be represented as the product of a unitary operator \(U\) with the operator of complex conjugation \( K \), i.e. \(A=UK\). We have \(A^{-1}=KU^{\dag }\) and \(Ki=-iK\).
- 23.
Only recently, the violation of time-reversal symmetry was detected directly for the first time (CPT invariance, however, remains valid); see J.P. Lee et al., ‘Observation of Time-Reversal Violation in the \(B^{0}\) Meson System’, Phys. Rev. Lett. 109, 211801 (2012). In this context, entirely new concepts, such as that of the ‘time crystal’, are emerging; cf. F. Wilczek, ‘Quantum Time Crystal’, Phys. Rev. Let. 109, 160401 (2012). Along with David Gross and David Politzer, Frank Wilczek was awarded the Nobel Prize for Physics in 2004 for the discovery of asymptotic freedom in the theory of strong interactions.
- 24.
More on vector operators in Appendix G, Vol. 2.
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Pade, J. (2014). Symmetries and Conservation Laws. In: Quantum Mechanics for Pedestrians 2: Applications and Extensions. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-00813-4_21
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