College Math Teaching

August 17, 2011

Quantum Mechanics and Undergraduate Mathematics XIV: bras, kets and all that (Dirac notation)

Filed under: advanced mathematics, applied mathematics, linear albegra, physics, quantum mechanics, science — collegemathteaching @ 11:29 pm

Up to now, I’ve used mathematical notation for state vectors, inner products and operators. However, physicists use something called “Dirac” notation (“bras” and “kets”) which we will now discuss.

Recall: our vectors are integrable functions \psi: R^1 \rightarrow C^1 where \int^{-\infty}_{\infty} \overline{\psi} \psi dx converges.

Our inner product is: \langle \phi, \psi \rangle = \int^{-\infty}_{\infty} \overline{\phi} \psi dx

Here is the Dirac notation version of this:
A “ket” can be thought of as the vector \langle , \psi \rangle . Of course, there is an easy vector space isomorphism (Hilbert space isomorphism really) between the vector space of state vectors and kets given by \Theta_k \psi = \langle,\psi \rangle . The kets are denoted by |\psi \rangle .
Similarly there are the “bra” vectors which are “dual” to the “kets”; these are denoted by \langle \phi | and the vector space isomorphism is given by \Theta_b \psi = \langle,\overline{\psi} | . I chose this isomorphism because in the bra vector space, a \langle\alpha,| =  \langle \overline{a} \alpha,| . Then there is a vector space isomorphism between the bras and the kets given by \langle \psi | \rightarrow |\overline{\psi} \rangle .

Now \langle \psi | \phi \rangle is the inner product; that is \langle \psi | \phi \rangle = \int^{\infty}_{-\infty} \overline{\psi}\phi dx

By convention: if A is a linear operator, \langle \psi,|A = \langle A(\psi)| and A |\psi \rangle = |A(\psi) \rangle Now if A is a Hermitian operator (the ones that correspond to observables are), then there is no ambiguity in writing \langle \psi | A | \phi \rangle .

This leads to the following: let A be an operator corresponding to an observable with eigenvectors \alpha_i and eigenvalues a_i . Let \psi be a state vector.
Then \psi = \sum_i \langle \alpha_i|\psi \rangle \alpha_i and if Y is a random variable corresponding to the observed value of A , then P(Y = a_k) = |\langle \alpha_k | \psi \rangle |^2 and the expectation E(A) = \langle \psi | A | \psi \rangle .


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