Math Help - Quantum 3

1. Quantum 3

1.) In Quantum, how are $\lambda, h,$ $p$ related to each other?
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I was able to do this.

We know the following: $k=\frac{2\pi}{\lambda}, \lambda = \frac{h}{p} \implies p = \frac{h}{\lambda}$. Hence, we're able to find $p$.

$\frac{\frac{h}{2\pi}}{\frac{\lambda}{2\pi}} = \frac{h}{\lambda} = \frac{h}{\frac{h}{p}} = p$

2.) Describe the physical meaning of $\psi(x,x_0,0) = \int_{-\infty}^{\infty}dk a(k,k_0)e^{ik(x-x_0)}$

which is for any wave function.
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The book describes this one as being a Fourier transform $a(k,k_0)$ and inverting the fourier integral.

Does this move position in momentum?

(On a side note, why don't physics people place the dk, dx etc AFTER the integral!).

3.) Describe the physical meaning of $\Delta x\Delta p \geq \bar{h}$. Why are the uncertainties in $x$ and $p$ intrinsic rather than being a type of measurement errors.
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Well, I don't understand the intrinsic part. The first part I took straight from my book:

The product of the position and momentum uncertainties $\Delta x$ and $\Delta p$, respectively, of a quantum object is greater than or equal to Planck's constant $\bar{h}$

Not sure what the second part is asking.

4.) Using $\Delta x\Delta p \geq \bar{h}$ (that is the uncertainty relation), determine (estimating) the minimum amount of energy of a particle with mass $m$ and with potential energy $V(x) = \frac{-k}{x}$, where $k$ is a pos. constant.
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Not sure.

2. Originally Posted by DiscreteW
1.) In Quantum, how are $\lambda, h,$ $p$ related to each other?
---------------------

I was able to do this.

We know the following: $k=\frac{2\pi}{\lambda}, \lambda = \frac{h}{p} \implies p = \frac{h}{\lambda}$. Hence, we're able to find $p$.

$\frac{\frac{h}{2\pi}}{\frac{\lambda}{2\pi}} = \frac{h}{\lambda} = \frac{h}{\frac{h}{p}} = p$
Right.

Originally Posted by DiscreteW
2.) Describe the physical meaning of $\psi(x,x_0,0) = \int_{-\infty}^{\infty}dk a(k,k_0)e^{ik(x-x_0)}$

which is for any wave function.
--------------------

The book describes this one as being a Fourier transform $a(k,k_0)$ and inverting the fourier integral.

Does this move position in momentum?

(On a side note, why don't physics people place the dk, dx etc AFTER the integral!).
Yes, this is an F transform of the function a. A Fourier transform "switches" the independent variable to its conjugate. Position and momentum are conjugate pairs.

I really don't know why the dx's etc. are written in front of the integrand. But I find that it makes the equations easier to read, personally.

Originally Posted by DiscreteW
3.) Describe the physical meaning of $\Delta x\Delta p \geq \bar{h}$. Why are the uncertainties in $x$ and $p$ intrinsic rather than being a type of measurement errors.
---------------------

Well, I don't understand the intrinsic part. The first part I took straight from my book:

The product of the position and momentum uncertainties $\Delta x$ and $\Delta p$, respectively, of a quantum object is greater than or equal to Planck's constant $\bar{h}$

Not sure what the second part is asking.
The uncertainty of an object's position is directly related to the fact that the object is being treated as a "wavefunction." A wavefunction is not localized in space. All we can do is find the region of maximum probability of the particle being there, and call this the position. Since this region is never a point, there must be some intrinsic "smearing" of the position of the particle. The same argument holds for the momentum. (There is a way to argue this in the "bra-ket" formalism, but I'll be if I understand it myself.)

Originally Posted by DiscreteW
4.) Using $\Delta x\Delta p \geq \bar{h}$ (that is the uncertainty relation), determine (estimating) the minimum amount of energy of a particle with mass $m$ and with potential energy $V(x) = \frac{-k}{x}$, where $k$ is a pos. constant.
-----------------------

Not sure.
Hint: The minimum energy is going to be for T = 0. Now note that
$\Delta V \approx \frac{d}{dx} \left ( \frac{-k}{x} \right ) \Delta x$

By the way, here's a Latex tip. The command for h-bar is "\hbar" inside the math brackets. $\hbar$

-Dan