Problem 4.1 – Griffith’s Intro to QM

Problem 4.1

(a) Work out all of the canonical commutation relations for components of the operators r and p: $[x,y]$ , $[x,p_y]$ , $[x,p_x]$ , $[p_y,p_z]$ , and so on.
Answer:

$[r_i,p_j]= -[p_i,r_j]=i\hbar \delta_{ij} \text{, } [r_i,r_j] = [p_i,p_j]=0,$

where the indices stand for $x$ , $y$ , or $z$ , and $r_x=x$ , $r_y=y$ , and $r_z=z$ .

(b) Confirm the three-dimensional version of Ehrenfest’s theorem,

$\dfrac{d}{dt} \langle \vec{r} \rangle = \dfrac{1}{m} \langle \vec{p} \rangle \text{, and } \dfrac{d}{dt} \langle \vec{p} \rangle = \langle \nabla V \rangle .$

(Each of these, of course, stands for three equations—one for each component.) Hint: First check that the “generalized” Ehrenfest theorem, Equation 3.73, is valid in three dimensions.

$\sigma_x\sigma_{p_x} \geq \hbar /2 \text{, } \sigma_y\sigma_{p_y} \geq \hbar /2 \text{, } \sigma_z\sigma_{p_z} \geq \hbar / 2$

but there is no restriction on, say, $\sigma_x\sigma_{p_y}$ .

Solution:

Problem 4.1 Solution (Download)

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Problem 4.1 – Griffith’s Intro to QM

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Problem 4.1 – Griffith’s Intro to QM

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