Symmetries

Hibert Space Review

  • $(v,w) \geq 0$, where (a,b) denotes an inner product
    • This is linear in the right argument, and anti-linear in the first argument (ie. take complex conjugates)
    • (v,v) = 0 implies v = 0
  • $v = \Sigma_{i} e_{i} (e_{i}, v)$ always holds in Hilbert spaces (ie. can always decompose into an eigenbasis, where $(e_{i}, e_{j}) = \delta_{ij}$)
    • This can be extended to non-orthonormal bases (ie. $(e_{i}, e_{j}) = G_{ij}$), which implies that $(v,w) = \Sigma_{ij} (v_{i}e_{i})G_{ij}^{-1} (e_{j}, w)$
  • You can add Hilbert spaces of different dimensions together
    • The new Hilbert space has the equivalence relationship of $<v_{1}+v_{2},w_{1}+w_{2}> = <v_{1},w_{1}> + <v_{2},w_{2}>$
    • The total dimensionality of the new Hilbert space is just the sum of the previous two
  • You can also multiply two Hilbert spaces together
    • The equivalence relation which must hold for the product is that $\lambda <v,w> = <\lambda v,w> = <v,\lambda w>$
    • You can decompose this product Hilbert space as a sum of tensor products of the eigenbases of each prior Hilbert space (ie. $H_{prod} = \Sigma_{ia} \Phi_{ia} e_{i} \otimes e_{a}$)

Cannonical Transforms

  • We know that cannonical transformations are an important part of classical Hamiltonian mechanics. What is the analog in quantum?
    • We want the probability to be conserved under a cannonical transformation! (ie. $P(\Phi) = |(\Phi,\Phi)|^{2}$)
    • We also want to preserve linearity (ie. $(\phi_{1}+\phi_{2})’ = \phi_{1}+\phi_{2}$)
    • The only transformations which satisfy these properties are either Unitary or Anti-Unitary

Unitary

  • $U(a\phi+b\psi)= aU\phi + bU\psi$ (can applies either before or after linearity)
  • $UU^{\dag} = I$, where the adjoint of an operator is defined as $(\psi O^{\dag} \phi) = (\phi O \psi)$
  • Needs to be invertible
  • Operator O transforms like $U^{\dag}O U = O'$
    • transform inner product, do adjoint on the left, compare

Anti-Unitary

  • $U(a\phi+b\psi)= a^{*}U\phi + b^{*}U\psi$ (ie. must maintain anti-linearity)
  • $(U\phi, U\psi) = (\psi, \phi) = (\phi, \psi)^{*}$
  • Operators transform like $U^{-1} O U$

Parity

  • Defined as $\Pi^{\dag} \vec{x} \Pi = -\vec{x}$
  • We also define $\Pi* \Pi = 1$, which implies $\Pi = \Pi^{-1}$ and that the eigenstates are $\pm 1$
  • Suppose that $\phi$ is an eigenstate of $\Pi$. Then $(\phi x \phi) = 0$ (Apply parity definition, transfer parity operator to states, apply eigen equation $\Pi \psi = \epsilon \psi$), and then note that you get $-q =q $, which implies q=0
    • As a consequence of this, if you Hamiltonian commutes with parity, then you know that you can write you state as a sum of even and odd states

Time Reversal

  • Time reversal (ie. letting $t \rightarrow -t$) must be described by an anti-unitary operator (otherwise, you get a sea of negative energies)