Transformations and symmetries in quantum mechanics

These notes give a brief and basic introduction to some central aspects concerning transformations and symmetries in quantum mechanics. Examples discussed include translations in space and time, as well as rotations. Translations in space are also called spatial translations, and sometimes even just " translations " for short, with " spatial " left implicit. To introduce the concept, let us consider the simplest example of a single particle in one spatial dimension. The state |x is the eigenstate of the position operatorˆx with eigenvalue x: ˆ x|x = x|x. (1) The eigenstates ofˆx obey x|x = δ(x − x). Now consider the state exp(−iˆp x ∆x/)|x, wherê p x is the momentum operator and ∆x is some arbitrary spatial displacement. As the commutator [ˆ x, ˆ p x ] = i is just a c-number, the simplified version (39) of the Baker-Hausdorff theorem holds, which gives ˆ x exp (−iˆp x ∆x/)|x = exp (−iˆp x ∆x/) exp (+iˆp x ∆x/)ˆ x exp (−iˆp x ∆x/) ˆ x+∆x |x = exp (−iˆp x ∆x/)(x + ∆x)|x = (x + ∆x) exp (−iˆp x ∆x/)|x. (2) This shows that exp (−iˆp x ∆x/)|x is an eigenstate ofˆx with eigenvalue x + ∆x. Also, because the operator exp(−iˆp x ∆x/) is unitary, the appropriate normalization is preserved: x| exp (+iˆp x ∆x/) exp (−iˆp x ∆x/) ˆ I |x = δ(x − x) = δ(x − ∆x − (x − ∆x)). (3) Therefore we conclude that exp (−iˆp x ∆x/)|x = |x + ∆x. (4) Thus the operator exp (−iˆp x ∆x/) transforms |x into |x + ∆x, i.e. it effects 1 translations in space by ∆x. If the displacement ∆x is infinitesimal, i.e. ∆x → dx, the operator 1 Do not confuse the word effect, used here as a verb, with the verb affect. To effect means to produce, bring about, accomplish, make happen.