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
$$\bra{b^k}=\bra{a^k} \hat{U}^{\dagger}.$$
 
Not an expert in QM. AFAIK, Schrödinger's equation is quite different from the classical wave equation. The former is an equation for the dynamics of the state of a (quantum?) system, the latter is an equation for the dynamics of a (classical) degree of freedom. As a matter of fact, Schrödinger's equation is first order in time derivatives, while the classical wave equation is second order. But, AFAIK, Schrödinger's equation is a wave equation; only its interpretation makes it non-classical...
I am not sure if this falls under classical physics or quantum physics or somewhere else (so feel free to put it in the right section), but is there any micro state of the universe one can think of which if evolved under the current laws of nature, inevitably results in outcomes such as a table levitating? That example is just a random one I decided to choose but I'm really asking about any event that would seem like a "miracle" to the ordinary person (i.e. any event that doesn't seem to...
Back
Top