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There is another consequence of the uncertainty principle for energy and time. If energy is uncertain by Δ E size 12{ΔE} {} , then conservation of energy can be violated by Δ E size 12{ΔE} {} for a time Δ t size 12{Δt} {} . Neither the physicist nor nature can tell that conservation of energy has been violated, if the violation is temporary and smaller than the uncertainty in energy. While this sounds innocuous enough, we shall see in later chapters that it allows the temporary creation of matter from nothing and has implications for how nature transmits forces over very small distances.

Finally, note that in the discussion of particles and waves, we have stated that individual measurements produce precise or particle-like results. A definite position is determined each time we observe an electron, for example. But repeated measurements produce a spread in values consistent with wave characteristics. The great theoretical physicist Richard Feynman (1918–1988) commented, “What there are, are particles.” When you observe enough of them, they distribute themselves as you would expect for a wave phenomenon. However, what there are as they travel we cannot tell because, when we do try to measure, we affect the traveling.

Section summary

  • Matter is found to have the same interference characteristics as any other wave.
  • There is now a probability distribution for the location of a particle rather than a definite position.
  • Another consequence of the wave character of all particles is the Heisenberg uncertainty principle, which limits the precision with which certain physical quantities can be known simultaneously. For position and momentum, the uncertainty principle is Δ x Δ p h size 12{ΔxΔp>= { {h} over {4π} } } {} , where Δ x size 12{Δx} {} is the uncertainty in position and Δ p size 12{Δp} {} is the uncertainty in momentum.
  • For energy and time, the uncertainty principle is Δ E Δ t h size 12{ΔEΔt>= { {h} over {4π} } } {} where Δ E size 12{ΔE} {} is the uncertainty in energy and Δ t size 12{Δt} {} is the uncertainty in time.
  • These small limits are fundamentally important on the quantum-mechanical scale.

Conceptual questions

What is the Heisenberg uncertainty principle? Does it place limits on what can be known?

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Problems&Exercises

(a) If the position of an electron in a membrane is measured to an accuracy of 1 . 00 μm size 12{1 "." "00 μm"} {} , what is the electron’s minimum uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV? (c) What are the implications of this energy, comparing it to typical molecular binding energies?

(a) 57.9 m/s

(b) 9 . 55 × 10 9 eV size 12{9 "." "55" times "10" rSup { size 8{ - 9} } " eV"} {}

(c) From [link] , we see that typical molecular binding energies range from about 1eV to 10 eV, therefore the result in part (b) is approximately 9 orders of magnitude smaller than typical molecular binding energies.

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(a) If the position of a chlorine ion in a membrane is measured to an accuracy of 1 . 00 μm size 12{1 "." "00 μm"} {} , what is its minimum uncertainty in velocity, given its mass is 5 . 86 × 10 26 kg size 12{5 "." "86" times "10" rSup { size 8{ - "26"} } " kg"} {} ? (b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?

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Suppose the velocity of an electron in an atom is known to an accuracy of 2 . 0 × 10 3 m/s size 12{2 "." 0 times "10" rSup { size 8{3} } " m/s"} {} (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

29 nm,

290 times greater

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The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

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A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?

1 . 10 × 10 13 eV size 12{1 "." "10" times "10" rSup { size 8{ - "13"} } " eV"} {}

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(a) The lifetime of a highly unstable nucleus is 10 20 s size 12{"10" rSup { size 8{ - "20"} } " s"} {} . What is the smallest uncertainty in its decay energy? (b) Compare this with the rest energy of an electron.

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The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?

3 . 3 × 10 22 s size 12{3 "." 3 times "10" rSup { size 8{ - "22"} } " s"} {}

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The decay energy of a short-lived nuclear excited state has an uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime it can have?

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What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?

2.66 × 10 46 kg size 12{2 "." "66" times "10" rSup { size 8{ - "46"} } " kg"} {}

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Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, Δ E Δ t h size 12{ΔE Δt approx h} {} , using the following arguments: Since the position of a particle is uncertain by Δ x λ size 12{Δx approx λ} {} , where λ size 12{λ} {} is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse Δ x size 12{Δx} {} . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to λ size 12{λ} {} . Find Δ t size 12{Δt} {} and Δ E size 12{ΔE} {} ; then multiply them to give the approximate uncertainty principle.

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Questions & Answers

explanations on harmonic motion
Tumwiine Reply
example ofchange of state of the body in the effectof heat
Abiodun Reply
what is normal force?
Neyaz Reply
the force that pushes upward on us. the force that opposes gravity
clifford
upthrust of air
Abdikadir
Newton's 3rd law. the force of the ground (earth) that pushes back on gravity, keeping us on the ground instead of sinking into it.
clifford
I really need lots of questions on frictional force
Ogboru Reply
Questions or answers?
Shii
I can help answering what I can
Shii
does friction also need some force to perform?
Mohit
no friction is a force just like the gravitational force
clifford
yeah but u can't apply friction anywhere else like other forces
Mohit
I don't understand that question. friction does work alongside other forces based on the situation.
clifford
eg. when walking there are two forces acting on us gravitational and frictional force. friction helps us move forward and gravity keeps us on the ground
clifford
friction is a contact force. Two surfaces are necessary for the force to work.
clifford
hope this helped
clifford
the friction force which oppose while it contact with surrounding. there are two kind of friction. slidding and rolling friction.
Neyaz
Two unequal masses M1 and M2 are connected by a string of tension T on a plane,find the acceleration and tension in the string
Ogboru
derive the equation
Ogboru
Hi
Olamide
What is physics?
Jeuloriz Reply
physics is a branch of science in which we are dealing with the knowledge of our physical things. macroscopic as well as microscopic. we are going look inside the univers with the help of physics. you can learn nature with the help of physics. so many branches of physics you have to learn physics.
vijay
What are quarks?
Breanna Reply
6 type of quarks
Neyaz
what is candela
Akani Reply
Candela is the unit for the measurement of light intensity.
Osei
any one can prove that 1hrpower= 746 watt
Neyaz Reply
Newton second is the unit of ...............?
Neyaz
Impulse and momentum
Fauzia
force×time and mass× velocity
vijay
Good
Neyaz
What is the simple harmonic motion?
Fauzia Reply
oscillatory motion under a retarding force proportional to the amount of displacement from an equilibrium position
Yuri
Straight out of google, you could do that to, I suppose.
Yuri
*too
Yuri
ok
Fauzia
Oscillatory motion under a regarding force proportional to the amount of displacement from an equilibrium position
Neyaz
examples of work done by load of gravity
Maureen Reply
What is ehrenfest theorem?
Fauzia Reply
You can look it up, faster and more reliable answer.
Yuri
That isn't a question to ask on a forum and I also have no idea what that is.
Yuri
what is the work done by gravity on the load 87kj,11.684m,mass xkg[g=19m/s
Maureen
What is law of mass action?
Fauzia Reply
rate of chemical reactions is proportional to concentration of reactants ...
muhammad
ok thanks
Fauzia
what is lenses
Ndobe Reply
lenses are two types
Fauzia
concave and convex
muhammad
right
Fauzia
speed of light in space
Vikash Reply
in vacuum speed of light is 3×10^8 m/s
vijay
ok
Vikash
2.99×10^8m/s
Umair
2.8820^8m/s
Muhammed
which is correct answer
Vikash
he is correct but we can round up in simple terms
vijay
3×10^8m/s
vijay
is it correct
Fauzia
I mean 3*10^8 m/s ok
vijay
299792458 meter per second
babar
3*10^8m/s
Neyaz
how many Maxwell relations in thermodynamics
vijay
how we can do prove them?
vijay
What is second law of thermodynamics?
Neyaz
please who has a detailed solution to the first two professional application questions under conservation of momentum
Kwaku Reply
I want to know more about pressure
Osei
I can help
Emeh
okay go on
True
I mean on pressure
Emeh
definition of Pressure
John
it is the force per unit area of a substance.S.I unit is Pascal 1pascal is defined as 1N acting on 1m² area i.e 1pa=1N/m²
Emeh
pls explain Doppler effect
Emmex
Practice Key Terms 6

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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