Three quantum mechanics questions (about Uncertainty p. and comp. conjugate)

[schniefen]

Homework Statement

See attached images.

Relevant Equations

$\Delta p \Delta x \geq \frac{h}{4\pi}$

Is the last inequality correct? Should it not be $|A|^2 \cdot 2(1+\cos{(ka)})$? How is the time calculated here? Given $\Delta v > 10^{-34}$... How come $mv \Delta v = \Delta (\frac{mv^2}{2})$? Where does the $(1/2)$ come from?

 

[BvU]

1. You are correct. Book solution is a mess
2. It's your homework: what's the relevant equation here ? (hint: see 3)
3. Simple differentiation

You should know by now PF homework fora require posted efforts, but I'm convinced you did your part.

 

[PeroK]

Homework Statement: See attached images.
Homework Equations: ΔpΔx≥h4πΔpΔx≥h4π

View attachment 252715 Is the last inequality correct? Should it not be |A|2⋅2(1+cos(ka))|A|2⋅2(1+cos⁡(ka))?View attachment 252716 How is the time calculated here? Given Δv>10−34Δv>10−34...View attachment 252717 How come mvΔv=Δ(mv22)mvΔv=Δ(mv22)? Where does the (1/2)(1/2) come from?

I must be honest, I don't buy this "uncertainly in the passage of time". What on Earth does that mean?

The Δt in the time-energy uncertainty relation is the time it takes the system to change in some defined way.

In QM, time is a parameter about which there is no uncertainty as such.

 

[BvU]

It's an 'order of magnitude' exercise

 

[schniefen]

Regarding 3, $mv \Delta v=\Delta (mv^2)$, but where does the last equality follow from?

 

[PeroK]

Regarding 3, $mv \Delta v=\Delta (mv^2)$, but where does the last equality follow from?

If you extend that notion of a differential to a derivative, with respect to time, then all should be clear.

 

[schniefen]

Why would $\Delta t > \frac{4\pi(\Delta x)^2}{h}$ give the time it takes for $\Delta x$ to double? Where is the factor $2\Delta x$?

 

[PeroK]

Why would $\Delta t > \frac{4\pi(\Delta x)^2}{h}$ give the time it takes for $\Delta x$ to double? Where is the factor $2\Delta x$?

The logic is that the uncertainty in $x$ has an associated minimum uncertainty in momentum, hence energy; and you apply the energy-time uncertainty relation.

Have you tried that?

 

[PeroK]

Why would $\Delta t > \frac{4\pi(\Delta x)^2}{h}$ give the time it takes for $\Delta x$ to double? Where is the factor $2\Delta x$?

The logic is this. You start with an uncertainty in $x$ in the classical sense. You apply the HUP to get an uncertainty in $v$. You interpret this as a classical uncertainty. You multiply the uncertainty in $v$ by $\Delta t$ to get a further, classical, uncertainty in $x$.

You want to find $\Delta t$ where the additional uncertainty in $x$ equals the initial uncertainty.

That looks nothing like the QM I know. What book is that?