Heisenberg Uncertainty Principle & particle

Smythe

By considering the representation of a particle by a series of plane waves, show how a similar expression to the Heisenberg Uncertainty Principle can be obtained relating energy and time.

I have attached the provided solution, and also below that my attempt.

The provided solution has E= p^2 / 2m but they don't appear to have carried the 2 in the denominator through the rest of the calculation.

Morbert

Smythe wrote: »

By considering the representation of a particle by a series of plane waves, show how a similar expression to the Heisenberg Uncertainty Principle can be obtained relating energy and time.

I have attached the provided solution, and also below that my attempt.

The provided solution has E= p^2 / 2m but they don't appear to have carried the 2 in the denominator through the rest of the calculation.

They are using the chain rule. Here is a similar approach:

[latex]E = \frac{p^2}{2m}[/latex]
[latex]\frac{dE}{dp} = \frac{2p}{2m} = \frac{p}{m}[/latex]
[latex]dE = \frac{p}{m}dp[/latex]

Smythe

Morbert wrote: »

They are using the chain rule. Here is a similar approach:

[latex]E = \frac{p^2}{2m}[/latex]
[latex]\frac{dE}{dp} = \frac{2p}{2m} = \frac{p}{m}[/latex]
[latex]dE = \frac{p}{m}dp[/latex]

Of course, thanks!

Lbeard

Morbert wrote: »

They are using the chain rule. Here is a similar approach:

Are they using the Chain Rule? [latex]\frac{1}{2m}[/latex]is a constant.

I'm confused as to why the statement is incorrect.

Is it abuse of operations in the proof?

Morbert

Lbeard wrote: »

I'm confused as to why the statement is incorrect.

What statement is incorrect?

Lbeard

Morbert wrote: »

What statement is incorrect?

I'll repeat the question here:

By considering the representation of a particle by a series of plane waves, show how a similar expression to the Heisenberg Uncertainty Principle can be obtained relating energy and time.
Solution:


[LATEX]E = \frac{p^{2}}{2m}[/LATEX]
[LATEX]t=\frac{m}{p}x[/LATEX]
[LATEX]\Delta E=\frac{p}{m}\Delta p[/LATEX]
[LATEX]\Delta t=\frac{m}{p}\Delta x[/LATEX]
[LATEX]\Delta E.\Delta t=\Delta p.\Delta x\geq \frac{\hbar}{2}[/LATEX]

But why is this incorrect?


Unless 'But why is this incorrect?', is Smythe's writing.

It looks correct to me. There are some rules governing that particular use of deltas, that I'm incredibly fuzzy on. [LATEX]\frac{\Delta A}{\Delta B}[/LATEX] is not simply a quotient. The numerator and denominator cannot just be moved around like a quotient - though I'm not sure what the rules are.

Morbert

Lbeard wrote: »

I'll repeat the question here:

By considering the representation of a particle by a series of plane waves, show how a similar expression to the Heisenberg Uncertainty Principle can be obtained relating energy and time.
Solution:


[LATEX]E = \frac{p^{2}}{2m}[/LATEX]
[LATEX]t=\frac{m}{p}x[/LATEX]
[LATEX]\Delta E=\frac{p}{m}\Delta p[/LATEX]
[LATEX]\Delta t=\frac{m}{p}\Delta x[/LATEX]
[LATEX]\Delta E.\Delta t=\Delta p.\Delta x\geq \frac{\hbar}{2}[/LATEX]

But why is this incorrect?


Unless 'But why is this incorrect?', is Smythe's writing.

It looks correct to me. There are some rules governing that particular use of deltas, that I'm incredibly fuzzy on. [LATEX]\frac{\Delta A}{\Delta B}[/LATEX] is not simply a quotient. The numerator and denominator cannot just be moved around like a quotient - though I'm not sure what the rules are.

Ah,

I assumed that was Smythe. I can't find anything incorrect about it.

Also yes, you have to be careful with the deltas, but in this case it's ok.

[edit] For posterity

http://en.wikipedia.org/wiki/Separation_of_variables

Smythe

Lbeard wrote: »

Unless 'But why is this incorrect?', is Smythe's writing.

Yes, that's mine.