# A Question while i study Modern Quantum physics by Sakurai

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1. Mar 3, 2017

In the book of Modern Quantum physics by Sakurai

I wonder how 1.6.26 can be 1.6.27.

2. Mar 3, 2017

### PeroK

Actually, I'll put hats on the operators (something that perhaps Sakurai ought to have thought about):

If you set $\mathbf{dx'} = (dx_1, 0, 0)$ then $\mathbf{\hat{K}} \cdot \mathbf{dx'} = \hat{K_1} dx_1$ and (1.6.26) becomes:

$-i \mathbf{\hat{x}}\hat{K_1} dx_1 + i\hat{K_1}\mathbf{\hat{x}}dx_1 = (dx_1, 0, 0)$

Where $dx_1$ is an "infinitesimal" number and can be cancelled, and $\mathbf{\hat{x}} = (\hat{x_1}, \hat{x_2}, \hat{x_3})$; giving:

$-i (\hat{x_1}\hat{K_1} - \hat{K_1}\hat{x_1}, \ \hat{x_2}\hat{K_1} - \hat{K_1}\hat{x_2},\ \hat{x_3}\hat{K_1} - \hat{K_1}\hat{x_3}) = (1, 0, 0)$

Which gives the result for $\hat{K_1}$. Repeat for $\mathbf{dx'} = (0, dx_2, 0)$ etc.

3. Mar 3, 2017

### BvU

Thanks, @PeroK. Out of curiosity: How did you know ${\bf x K} \cdot d{\bf x'}$ was to be interpreted as ${\bf x} \left ({\bf K} \cdot d{\bf x'} \right)$ and not as $\left ({\bf x K} \right ) \cdot d{\bf x'}$ ?

4. Mar 3, 2017

### PeroK

I looked at my notes and, unlike Sakurai, I tend to distinguish between operators and numbers/vectors. You have to do that at the beginning of the section/proof and keep track of it.