I will present a general rule for multiplying a quantum Schubert polynomial by a quantum power sum polynomial. This is achieved by relating the structure constants for the multiplication of a quantum Schubert polynomial by a hook quantum Schur polynomial with the structure constants for the (classical) multiplication of a Schubert polynomial by a hook Schur polynomial. This is joint work with C Benedetti, N Bergeron, L Colmenarejo and F Sottile.