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

• schniefen
In summary, the conversation discusses the correctness of an inequality and the calculation of time in relation to quantum mechanics. It also addresses the concept of uncertainty in time and how it relates to the Heisenberg uncertainty principle. The conversation includes a discussion on the application of classical and quantum concepts in determining time and uncertainty. Overall, the conversation highlights the complexity and varying approaches to understanding quantum mechanics.
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?

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.

schniefen
schniefen said:
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.

It's an 'order of magnitude' exercise

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

schniefen said:
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##?

schniefen said:
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?

schniefen said:
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?

## 1. What is the Uncertainty Principle in quantum mechanics?

The Uncertainty Principle, also known as Heisenberg's Uncertainty Principle, states that it is impossible to know the exact position and momentum of a quantum particle at the same time. This means that the more precisely we know the position of a particle, the less we know about its momentum, and vice versa.

## 2. What is the significance of the Uncertainty Principle?

The Uncertainty Principle has significant implications for our understanding of the behavior of particles at the quantum level. It shows that there are fundamental limits to how precisely we can measure certain properties of particles, and that certain properties cannot be known simultaneously.

## 3. What is the complementary relationship between position and momentum in quantum mechanics?

The position and momentum of a particle are complementary variables, meaning that they are related in such a way that the more precisely we know one, the less we know about the other. This is a fundamental concept in quantum mechanics and is described by Heisenberg's Uncertainty Principle.

## 4. What is conjugate in quantum mechanics?

In quantum mechanics, conjugate variables are pairs of physical properties that are related in such a way that if we know the value of one, we can determine the value of the other. Examples of conjugate variables include position and momentum, as well as energy and time.

## 5. How does the uncertainty principle relate to computational conjugate variables?

Computational conjugate variables are pairs of properties that are related in such a way that they cannot both be known precisely at the same time due to the limitations of computing power. This is similar to the Uncertainty Principle in quantum mechanics, where certain properties of particles cannot be known simultaneously due to the limitations of measurement.

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