Help with understanding the uncertainty principle

[sodper]

[SOLVED] Help with understanding the uncertainty principle

I may have posted this in the wrong forum. If so I am sure a moderator could move it to the correct one.

Problem statement:
I have an assignment in which I need to calculate the lowest possible mean of an electron's KE , <KE>, based on the uncertainty of the electron's moment.

Given data:
The radius in which the electron is allowed to move, d=2.81*10^-10

Attempt at solution:
The uncertainty of the electrons position corresponds to d, deltax=d

Using the uncertainty principle I got, deltap = hbar/(2*deltax), where deltap is the lowest possible deviation.

I got deltaKE from, deltaKE = deltap^2/(2m), (m is the electron mass)

A fellow student explained to me that because my problem involves a stationary system (an electron bound to an atom), <KE> should be zero, which gives me:

(deltaKE)^2 = <KE^2> - <KE>^2 => (deltaKE)^2 = <KE^2>

But how do I proceed from here?

 

[olgranpappy]

I may have posted this in the wrong forum. If so I am sure a moderator could move it to the correct one.

Problem statement:
I have an assignment in which I need to calculate the lowest possible mean of an electron's KE , <KE>, based on the uncertainty of the electron's moment.

Given data:
The radius in which the electron is allowed to move, d=2.81*10^-10

Attempt at solution:
The uncertainty of the electrons position corresponds to d, deltax=d

Using the uncertainty principle I got, deltap = hbar/(2*deltax), where deltap is the lowest possible deviation.

I got deltaKE from, deltaKE = deltap^2/(2m), (m is the electron mass)

A fellow student explained to me that because my problem involves a stationary system (an electron bound to an atom), <KE> should be zero,

Ah... the good ol' "fellow student". Unfortunately, the fellow student is often not correct. For a stationary state $\delta <H>$ is zero, not <KE> (where H=KE + PE). The way you were proceeding previously by considering delta p as related to the given delta x looks fine to me.

 

[olgranpappy]

for example, if the electron is confined to an atom of size 'a' then the KE is roughly
$$\frac{\hbar^2}{m a^2}$$
(c.f. particle in a box energy levels (\frac{\hbar^2 \pi^2}{2 m a^2}), etc).

 

[sodper]

I don't think I quite understand. Did you, as I did, derive the expression for the KE from KE = p^2/(2m) ?
I simply substituted p with deltap to get the min deviation in KE.

I'm not sure how I am supposed use the expression of H.

 

[olgranpappy]

I don't think I quite understand. Did you, as I did, derive the expression for the KE from KE = p^2/(2m) ?
I simply substituted p with deltap to get the min deviation in KE.

yeah. that's right.

I'm not sure how I am supposed use the expression of H.

dont use it for anything; I was just pointing out that in a stationary state it is the *total* energy which has no variance, not the *kinetic* energy.

 

[sodper]

Ok, but is the expression (deltaKE)^2 = <KE^2> valid?

If so, can I just take the square root to get the mean KE?

 

[olgranpappy]

Ok, but is the expression (deltaKE)^2 = <KE^2> valid?

no... but we are just getting rough estimates here. That's all that is required. That's why we are using the "equation"

KE "=" (delta p)^2/2m

where (delta p) is given by \hbar divided by (delta x).

 

[sodper]

I see. Thanks! You've made it a lot clearer.

I'll post again if I need more help.