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In quantum mechanics, if a family of unitary operators $\{U(\alpha)\}$ depending on a bunch of continuous parameters $\{\alpha\}$, commute with the Hamiltonian of a system, we say that the system has a continuous symmetry. I think that we define symmetry in this manner because we want the Schrodinger equation to remain unchanged after the symmetry transformation. I think this is true but I don't have any source to validate my impression. Please tell me if my reasoning below is correct.

Under the symmetry, $|\psi\rangle\to |\psi'\rangle=U|\psi\rangle$ and $H\to H'= UHU^{-1}=H$, then $$i\hbar\frac{\partial|\psi\rangle}{\partial t}=H|\psi\rangle\to i\hbar\frac{\partial|\psi'\rangle}{\partial t}=H|\psi'\rangle$$

Gert
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Solidification
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1 Answers1

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Yes, what you wrote is correct.

One advanced note I wish someone had told me explicitly when I was learning quantum mechanics: when you get to study more complex systems (and in particular more complex spacetime symmetries), sometimes it is useful to generalize the definition of symmetry from the one you have given. In particular, rather than equating the set of operators that commute with the Hamiltonian with the set of symmetries, sometimes it is useful to think of the Hamiltonian operator as being one of a number of operators in an algebra, with prescribed commutation relationships. For example, we want Galilean relativity to be a symmetry of non-relativistic quantum mechanics. It turns out that the operator which represents boosts is the position of the center of mass, $K=x+ p t / m$. This operator does not commute with the Hamiltonian, but the Hamiltonian and boost operator together are part of the Galilean algebra, which ensures that states form unitary representations of the Galilean group (which then ensures that the quantum theory has the Galilean symmetry). This is a more advanced notion of symmetry and may not be directly relevant for your work, but in my opinion it's at least useful to be aware that sometimes it is useful to generalize the idea that "symmetries must commute with the Hamiltonian".

Andrew
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  • Thanks @Andrew for the answer. But I did not quite comprehend what exactly is the condition for Galilean symmetry. Galilean algebra is a set of commutation relations. In what sense this is a symmetry? What does it leave unchanged? – Solidification Mar 04 '21 at 07:10
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    The more abstract notion of a symmetry comes from Wigner. Any linear operator on Hilbert space that leaves the absolute value of the inner product between two arbitrary states unchanged is a symmetry. This boils down to saying that unitary and anti-unitary operators represent symmetries. The magic is then that these symmetry operators can form representations of groups. Because of the properties of representations, you can use the symmetry to classify the states of the system. – Andrew Mar 04 '21 at 11:21
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    Groups are defined by a composition rule that tells you how to compute a third group element from two others. For continuous groups, you can (locally, ie for values of the continuous parameter that are close to the identify matrix) encode the information in the composition rule by the Lie algebra, or commutation relationships of "generators", operators that perform infinitesimally small group transformations. You can use the representation of the group to classify the states of the system. – Andrew Mar 04 '21 at 11:26
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    You may have run across the angular momentum operators. These are actually representations for the generators of the group $SO(3)$, which is the group corresponding to rotations in 3 dimensions. You can classify the states by the $\ell,m$ quantum numbers. In particle physics, to obey special relativity, you need to form representations of the Poincaire group; following the same logic leads to classifying particles by mass and spin. – Andrew Mar 04 '21 at 11:29
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    Again: I suspect you do not need all of this machinery right away. The notion of "operators commuting with the Hamiltonian" is easier and useful because the quantum numbers associated with those operators will not change with time. But it doesn't hurt to be exposed to more general formulations before you need them, so you can put an asterisk on your knowledge and come back to it if you need to. – Andrew Mar 04 '21 at 11:31
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    @mithusengupta123 I think saying that symmetries only exist when the corresponding generators commute with the Hamiltonian is too strong. For example boost operators are necessary in order to have Lorentz symmetry but boosts do not commute with the Hamiltonian. There is something special about commuting with the Hamiltonian, since generators which commute with the Hamiltonian are the ones which will have associated quantum numbers that do not depend on time. But, I would say that the most general approach is that symmetries are unitary representations of symmetry groups on Hilbert space. – Andrew Mar 04 '21 at 14:06
  • Having seen various other posts, I think I agree now that I was using (and thus confused by) a restrictive definition of symmetry. This answer by Moretti is also particularly useful: https://physics.stackexchange.com/questions/578290/the-definition-of-the-weaker-notion-of-symmetry-in-the-sense-of-wigners-theorem?noredirect=1&lq=1 But your approach to the meaning of symmetry will not lead to conservation laws. Am I wrong? – Solidification Mar 04 '21 at 14:10
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    @mithusengupta123 Exactly, conservation laws are only associated with operators that commute with the Hamiltonian, because this is the condition for a quantity being time independent. But the notion of a symmetry in quantum mechanics is more general, and not every generator in a symmetry group has an associated conserved quantity. – Andrew Mar 04 '21 at 14:19