A Matrix representation of a unitary operator, change of basis

Kashmir
Messages
466
Reaction score
74
If ##U## is an unitary operator written as the bra ket of two complete basis vectors :##U=\sum_{k}\left|b^{(k)}\right\rangle\left\langle a^{(k)}\right|##

##U^\dagger=\sum_{k}\left|a^{(k)}\right\rangle\left\langle b^{(k)}\right|##

And we've a general vector ##|\alpha\rangle## such that ##|\alpha\rangle=\sum_{a^{\prime}}\left|a^{\prime}\right\rangle\left\langle a^{\prime} \mid \alpha\right\rangle##

Sakurai writes at pg 50 :
"how can we obtain ##\left\langle b^{\prime} \mid \alpha\right\rangle##, the expansion coefficients in the new basis? answer is very simple: Just multiply (1.5.9) by ##\left\langle b^{(k)}\right|##
##
\left\langle b^{(k)} \mid \alpha\right\rangle=\sum_{l}\left\langle b^{(k)} \mid a^{(l)}\right\rangle\left\langle a^{(l)} \mid \alpha\right\rangle=\sum_{l}\left\langle a^{(k)}\left|U^{\dagger}\right| a^{(l)}\right\rangle\left\langle a^{(l)} \mid \alpha\right\rangle .
##
##(1.5 .1##
In matrix notation, (1.5.10) states that the column matrix for ##|\alpha\rangle## in the new basis can be obtained just by applying the square matrix ##U^{\dagger}## to the colum matrix in the old basis:
##\quad(\mathrm{New})=\left(U^{\dagger}\right)(##old ##)##"So if the matrix representing ##U^\dagger## is applied on to the matrix representing ##|\alpha\rangle## ,it gives the vectors representation in the new basis. But when I apply ##U^\dagger## onto say an basis vector ##\left|a_{1}\right\rangle## ,it doesn't give me the vectors representation in new basis as shown below :

##\begin{aligned} U^{\dagger}\left|a_{1}\right\rangle &=\sum_{k}\left|a^{k}\right\rangle\left\langle b^{k} \mid a_{1}\right\rangle \\ &=\sum_{k}\left(\left\langle b^{k} \mid a_{1}\right\rangle\right) \cdot\left|a^{k}\right\rangle \end{aligned}##
 
Physics news on Phys.org
Why should it? You rather have
$$\ket{b^k}=\hat{U} \ket{a^k}.$$
Sakurai in the quoted text uses the adjoint of this
$$\bra{b^k}=\bra{a^k} \hat{U}^{\dagger}.$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
If we release an electron around a positively charged sphere, the initial state of electron is a linear combination of Hydrogen-like states. According to quantum mechanics, evolution of time would not change this initial state because the potential is time independent. However, classically we expect the electron to collide with the sphere. So, it seems that the quantum and classics predict different behaviours!
Back
Top