# Definition The [[group]] of symmetry operations $\hat{P}_R$ of the quantum system which leave the [[Hamiltonian]] invariant are called the group of the [[Schrödinger equation]]. They [[Commutator|commute]] with the Hamiltonian $\mathcal{H}$. $ [\mathcal{H}, \hat{P}_R] = 0 $ To show that these operators form a group, consider the following: 1. *Closure:* the product of two operators is still an operator in the group since we can consider them as separately acting on the Hamiltonian. 2. *Associativity:* associativity must hold since these are physical operations. 3. *Identity*: the identity element must exist physically (i.e. leaving the system unchanged). 4. *Inverse:* Each symmetry operator $\hat{P}_R$ has an inverse $\hat{P}_R^{-1}$ to undo the operation, based on physical considerations. This element is also in the group. Note that, if $ \mathcal{H} \ket{\psi} = E \ket{\psi} $ then $ \mathcal{H} \hat{P}_R \ket{\psi} = E \hat{P}_R\ket{\psi} $ by virtue of commutativity. That is, these symmetry operations do not alter the spectrum of the Hamiltonian. # Irreducible Representations We have $ \hat{P}_g \psi_{n \alpha} = \sum_j D^{(n)}(g)_{j\alpha} \psi_{nj} $ where $\hat{P}_g$ is the symmetry operator corresponding to the [[group]] [[Group Element|element]] $g$ and $D$ is the matrix [[Group Representation|representation]] of $g$. The [[wavefunction|wavefunctions]] are labeled by $n$, the Hamiltonian [[eigenvalue]] $E_n$ and $\alpha$, the degeneracy index which represents the partners of the [[Irreducible Representation|irrep]]. We can show that these $D^{(n)}(g)$ matrices form an $\ell_n$ dimensional [[Irreducible Representation|irrep]] of the group of the Schödinger equation, where $\ell_n$ is the [[degeneracy]] of the energy eigenvalue $E_n$.