The uncertainty principle can be expressed as a relation between the uncertainty ΔE in the energy state of a system and the time interval Δt during which the system remains in that state. In symbols,

ΔEΔt≥ℏ/2,

where ℏ=h/2π, h is Planck's constant.

The energy-time uncertainty principle says that the longer a system remains in the same energy state, the higher the accuracy (or the smaller the uncertainty) a measurement of that energy can be. Another implication is that physical processes can violate the law of energy conservation as long as the violation occurs for only a short time, determined by the uncertainty principle. This idea is at the base of the theory of virtual particles.

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PART A

Consider two electrons that interact with each other. Classically, their interaction would be described in terms of the electrostatic force. In quantum mechanics, their interaction is interpreted in terms of emission and absorption of photons: One of the two electrons emits a photon with energy ΔE, which is then absorbed by the other electron after a short period of time.

How long can the photon survive before it is absorbed without violating the uncertainty principle?

Express your answer in terms of ΔE, and h or ℏ.

ANSWER:Δt =

ℏ/2ΔE

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PART B - MAIN PROBLEM - HELP!!!!!

The same concepts used to describe the emission and absorption of virtual photons in electromagnetic interactions can be used to describe nuclear interactions. According to this theory, when two protons of a nucleus interact with each other, a virtual particle is created. In order to effectively mediate the nuclear force, the virtual particle must exist long enough to travel a distance comparable to the size of the nucleus: a distance on the order of 1.5×10−15m.

Assuming that the virtual particle moves at a speed close to c, what is its minimum mass?

To simplify the calculation, assume that the kinetic energy of the particle is negligible compared to its rest energy. This assumption is not really valid for particles moving at close to the speed of light, but it will lead to an answer of a reasonable order of magnitude.

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Hint 1.

Find the time interval during which the virtual particle survives

If the particle has to travel at least 1.5⋅10^−15m, how long does it survive?

Δt = ??

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Hint 2.

Find the minimum necessary uncertainty in energy.

What is the minimum uncertainty in energy that is necessary to create such a particle?

(ΔE)min = ??

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Δm = ?? kg

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My Comments:

I'm really unsure how to even solve Part B even not knowing how to solve the two hints listed above.

How do I go about this, knowing the formula for delta time in the first Part, and also that the

distance travelled by the particle has to be at least 1.5⋅10^−15m during its survival, so what do I do to solve the two hints (uncertainty in time, uncertainty in E(min)) then go for solving the uncertainty in mass for Part B?

Thanks!