How Does Quantum Mechanics Describe Particle Dynamics in Spherical Coordinates?

[unscientific]

Homework Statement

Part (a): By writing L² in einstein notation, show that p² can be written as:

Part(b): Show $\vec{p}.\hat {\vec{r}} - \hat {\vec{r}}.\vec{p} = -2i\hbar\frac{1}{r}$

Part(c): Show $\vec{r}.\vec{p} = rp_r + i\hbar$

Part (d): Show $p^2 = p_r^2 + \frac{L}{r^2}$

I missed out terms in part (a), I couldn't get parts (b) and (c) of this question.

Homework Equations

The Attempt at a Solution

Part (a)

$$L^2 = \epsilon_{ijk}x_j p_k \epsilon_{ilm} x_l p_m$$
$$= \epsilon_{ijk}\epsilon{ilm} x_j p_k x_l p_m$$
$$= \left(\delta_{jl}\delta_{km} - \delta_{jm}\delta_{kl}\right)x_jp_kx_lp_m$$

$$= x_jp_kx_jp_k - x_jp_kx_kp_j$$
$$= x_j^2p_k^2 - x_jp_jp_kx_k$$
$$= x_j^2p_k^2 - x_jp_j$$
$$= r^2p^2 - (\vec{r} . \vec {p})$$


Part(b)

$$\vec{p}.\hat {\vec{r}} - \hat {\vec{r}}.\vec{p} = -i\hbar\left(\nabla . (\frac{\vec{r}}{r})\right) + i\hbar\left(\frac{\vec{r}}{r}.\nabla\right)$$

Now using product rule:

$$= -i\hbar\left[ \vec{r}.(\nabla \frac{1}{r}) + \frac{1}{r}\nabla . \vec{r}\right] + i\hbar\left[\frac{1}{r} \vec{r}.\nabla\right]$$

Now, $\nabla \frac{1}{r} = -\frac{1}{r^3}\vec{r}$ and $\nabla . \vec{r} = 3$. Using these results, we obtain:

$$= \frac{i\hbar}{r}$$

Part (c)

Using result of part (b):

$$\vec{r}.\vec{p} = 2i\hbar + r(\vec{p}.\hat{\vec{r}})$$
$$= 2i\hbar + \vec{p}.\vec{r}$$
$$= 2i\hbar -i\hbar\nabla . \vec{r}$$
$$= 2i\hbar - 3i\hbar$$
$$= -i\hbar$$

Part(d)

Chuck parts (b) and (c) into the equation and you get the answer.

The physical interpretation is:

Total Energy = Radial Kinetic Energy + Rotational Energy

 

[MisterX]

For part (a), I think you made a mistake on the 5th line. You aren't allowed to just swap the order of x's and p's.

 

[ChrisVer]

for part b, try acting with the difference on a function...and avoid playing algebraically just with the operators alone...
For example after using the chain rule, you would cancel out terms with having a function being acted on...

 

[unscientific]

For part (a), I think you made a mistake on the 5th line. You aren't allowed to just swap the order of x's and p's.

OK here's what I got:

$$x_jp_kx_jp_k - x_jp_kx_kp_j$$

Now i change $p_k$ in first and second term to $-i\hbar\partial_k$:

$$-i\hbar x_j\partial_k(x_jp_k) +i\hbar x_j\partial_k(x_kp_j)$$

Using product rule: $\partial_kx_j = \delta_{kj}$

$$i\hbar \left[-x_jp_k\delta_{kj} - x_j^2\partial_kp_k + x_jp_j + x_jx_k\partial_kp_j\right]$$

Converting $\partial_k = \frac{i}{\hbar}p_k$:

$$i\hbar \left[ -x_jp_j - \frac{i}{\hbar} x_j^2p_k^2 + x_jp_j +\frac{i}{\hbar}x_jx_kp_kp_j\right]$$

 

[ChrisVer]

you should use the commutation relations for exchanging the x and p's...So in general:
$p_{i} x_{j}= -i \hbar δ_{ij} + x_{j} p_{i}$

it's still not trivial to write the momentum as the derivative and making it act on x alone... you are still having operators and as such they act on functions ... I'll repeat, avoid treating them as functions alone and work algebraically so "free".

For example take the commutator of p and x:
$[p_{i},x_{j}] = p_{i}x_{j} - x_{j} p_{i} = -i\hbar δ_{ij} + x_{j} i\hbar \frac{d}{dx_{i}}$
is not correct

Instead you take:

$[p_{i},x_{j}] f= p_{i}x_{j}f - x_{j} p_{i}f= -i\hbar \frac{d}{dx_{i}}(x_{j}f)+ i\hbar x_{j} \frac{d}{dx_{i}}f = -i\hbar δ_{ij} f$
so you take
$[p_{i},x_{j}]= p_{i}x_{j} - x_{j} p_{i} =-i\hbar δ_{ij}$

 

[unscientific]

you should use the commutation relations for exchanging the x and p's...So in general:
$p_{i} x_{j}= i \hbar δ_{ij} + x_{j} p_{i}$
(I may forget a - in front of $i \hbar$ )

it's still not trivial to write the momentum as the derivative and making it act on x alone... you are still having operators and as such they act on functions ... I'll repeat, avoid treating them as functions alone and work algebraically so "free"

That's exactly the relation I was looking for. thanks a lot

 

[ChrisVer]

That's exactly the relation I was looking for. thanks a lot

it's basic (before the central potential you are trying to work with)

 

[unscientific]

it's basic (before the central potential you are trying to work with)

We need to show this: $x_j^2p_k^2 - (x_jp_j)(x_kp_k) +i\hbar(x_jp_j)$

Starting:

$$x_jp_kx_jp_k - x_jp_kx_kp_j$$

Using $[x_i,p_j] = x_ip_j - p_jx_i = i\hbar \delta_{ij}$:

$$x_j(x_jp_k - i\hbar \delta_{jk})p_k - x_j(x_kp_k - i\hbar)p_j$$
$$x_j^2p_k^2 - i\hbar x_jp_j - x_j(x_kp_k)p_j + i\hbar x_jp_j$$

Using $[x_kp_k,p_j] = x_kp_kp_j - p_jx_kp_k = i\hbar \delta_{jk}p_k$

$$= x_j^2p_k^2 - i\hbar x_jp_j - x_j(i\hbar\delta_{jk}p_k + p_jx_kp_k) + i\hbar x_jp_j$$

$$= x_j^2p_k^2 - i\hbar x_jp_j - x_jp_jx_kp_k$$
$$= r^2p^2 - i\hbar (\vec{r}.\vec{p}) - (\vec{r}.\vec{p})(\vec{r}.\vec{p})$$

which is different from the answer..

 

[unscientific]

bumpp

 

[unscientific]

bumppp

 

[unscientific]

bumpp