1.) In Quantum, how are $\displaystyle \lambda, h,$ $\displaystyle p$ related to each other?

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I was able to do this.

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

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

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

which is for any wave function.

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The book describes this one as being a Fourier transform $\displaystyle a(k,k_0)$ 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 $\displaystyle \Delta x\Delta p \geq \bar{h}$. Why are the uncertainties in $\displaystyle x$ and $\displaystyle p$ 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 $\displaystyle \Delta x$ and $\displaystyle \Delta p$, respectively, of a quantum object is greater than or equal to Planck's constant $\displaystyle \bar{h}$

Not sure what the second part is asking.

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

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Not sure.