Heisenberg Uncertainty Principle - Revision history

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History

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	==History==		==History==

	Quantum theory was first developed in the mid 1920s, with an emphasis on the observable properties of quantum systems. Heisenberg, guided by experimental results ~~putting~~ the product of certain uncertainties around <math> {\hbar} </math>~~, he~~ linked it to the mathematical formalism of linear operators. Namely, he found the commutator of certain operators to be <math> {i\hbar} </math>. He reasoned that since simple measurements of wavelength, time, and position can only be done by interaction, the magnitude of such interactions must affect future measurements. For example, observing an electron amounts to sending or receiving a photon from it. However, doing this alters the electron.		Quantum theory was first developed in the mid 1920s, with an emphasis on the observable properties of quantum systems. Heisenberg was guided by experimental results suggesting that the product of certain uncertainties around <math> {\hbar} </math>. He linked this to the mathematical formalism of linear operators. Namely, he and his contemporaries found the commutator of certain operators to be <math> {i\hbar} </math>. He reasoned that since simple measurements of wavelength, time, and position can only be done through interaction, the magnitude of such interactions must affect future measurements. For example, observing an electron amounts to sending or receiving a photon from it. However, doing this alters the electron.

	===Single-slit Diffraction Experiments===		===Single-slit Diffraction Experiments===

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Middling

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	where <math> {\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}} </math> (see [[Lorentz Transformations]]). So the muon's momentum is <math> {1.15\times 10^{-20}} </math> kg m/s. Taking <math> {10\% p=\Delta p=1.15\times 10^{-21}} </math> kg m/s, the minimum uncertainty in position is		where <math> {\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}} </math> (see [[Lorentz Transformations]]). So the muon's momentum is <math> {1.15\times 10^{-20}} </math> kg m/s. Taking <math> {10\% p=\Delta p=1.15\times 10^{-21}} </math> kg m/s, the minimum uncertainty in position is

	<math> {\Delta x \geq \frac{\hbar}{2\Delta p}=4.57 \times 10^{-15}} </math> meters.		<math> {\Delta x \geq \frac{\hbar}{2\Delta p}=4.585 \times 10^{-14}} </math> meters.

	===Difficult===		===Difficult===

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	[[File:Positionvector.jpg\|border\|caption]]		[[File:Positionvector.jpg\|border\|caption]]

	An actual quantum state vector actually takes complex values for its components. It would be more accurate to add another dimension to both of the above graphs, indicating the imaginary part of each component. This has been omitted for simplicity.		An actual quantum state vector actually takes complex values for its components. It would be more accurate to add another dimension to both of the above graphs, indicating the imaginary part of each component. This has been omitted for simplicity. See [[The Born Rule]] for a more rigorous treatment of quantum state vectors.

	The measurement of some property amounts to acting on this vector with an "operator". Every measurable quantity, like position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by		The measurement of some property amounts to acting on this vector with an "operator". Every measurable quantity, like position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by
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	==History==		==History==

	~~The development~~ of the ~~Heisenberg uncertainty principle is largely accredited~~ to ~~Werner Heisenberg~~.e		Quantum theory was first developed in the mid 1920s, with an emphasis on the observable properties of quantum systems. Heisenberg, guided by experimental results putting the product of certain uncertainties around <math> {\hbar} </math>, he linked it to the mathematical formalism of linear operators. Namely, he found the commutator of certain operators to be <math> {i\hbar} </math>. He reasoned that since simple measurements of wavelength, time, and position can only be done by interaction, the magnitude of such interactions must affect future measurements. For example, observing an electron amounts to sending or receiving a photon from it. However, doing this alters the electron.

	===Single-slit Diffraction Experiments===		===Single-slit Diffraction Experiments===
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	Libretexts. (2021, April 23). Heisenberg's Uncertainty Principle. Chemistry LibreTexts. Retrieved April 20, 2022, from chem.libretexts.org		Libretexts. (2021, April 23). Heisenberg's Uncertainty Principle. Chemistry LibreTexts. Retrieved April 20, 2022, from chem.libretexts.org

			Hilgevoord, Jan and Jos Uffink, "The Uncertainty Principle", The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/win2016/entries/qt-uncertainty/>.


	[[Category:Which Category did you place this in?]]		[[Category:Which Category did you place this in?]]

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	==Examples==		==Examples==

	~~TODO: ADD EXAMPLES~~

	~~Be sure to show all steps in your solution and include diagrams whenever possible~~

	===Simple===		===Simple===
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	===Difficult===		===Difficult===

	An emitted photon's wavelength has been determined to be 530 nm within a range of 2 nm. What is the uncertainty of the time of emission~~, assuming it is the same magnitude of the time of emission.~~		An emitted photon's wavelength has been determined to be 530 nm within a range of 2 nm. What is the uncertainty of the time of emission?

	Solution: A photon's wavelength is related to its energy by <math> {E=c/\lambda} </math>. Uncertainty in energy and time are related by the Uncertainty principle, <math> {\Delta E \Delta t \geq \frac{\hbar}{2}} </math>. Uncertainties are relatively small quantities, making them analogous to differentials. So another relationship between wavelength and energy can be found:		Solution: A photon's wavelength is related to its energy by <math> {E=c/\lambda} </math>. Uncertainty in energy and time are related by the Uncertainty principle, <math> {\Delta E \Delta t \geq \frac{\hbar}{2}} </math>. Uncertainties are relatively small quantities, making them analogous to differentials. So another relationship between wavelength and energy can be found:

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problems

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 Solution: The minimum uncertainties are related by <math> {\Delta x \Delta p \geq \hbar/2} </math>. Taking the uncertainty of the particle's position to be the width of the box, the uncertainty of the momentum is then
-<math> {\Delta p \geq \frac{\hbar}{2\Delta x}\approx 5.27\times 10^-25} </math> kg m/s.
+<math> {\Delta p \geq \frac{\hbar}{2\Delta x}\approx 5.27\times 10^{-25}} </math> kg m/s.
 ===Middling===
 ===Difficult===
 ==Connectedness==

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	Solution: The minimum uncertainties are related by <math> {\Delta x \Delta p \geq \hbar/2} </math>. Taking the uncertainty of the particle's position to be the width of the box, the uncertainty of the momentum is then		Solution: The minimum uncertainties are related by <math> {\Delta x \Delta p \geq \hbar/2} </math>. Taking the uncertainty of the particle's position to be the width of the box, the uncertainty of the momentum is then

	<math> {\Delta p \geq \frac{\hbar}{2\Delta x}\approx 5.27\times 10^-25} </~~mat~~> kg m/s.		<math> {\Delta p \geq \frac{\hbar}{2\Delta x}\approx 5.27\times 10^-25} </math> kg m/s.

	===Middling===		===Middling===

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	An actual quantum state vector actually takes complex values for its components. It would be more accurate to add another dimension to both of the above graphs, indicating the imaginary part of each component. This has been omitted for simplicity.		An actual quantum state vector actually takes complex values for its components. It would be more accurate to add another dimension to both of the above graphs, indicating the imaginary part of each component. This has been omitted for simplicity.

	The measurement amounts to acting on this vector with an "operator". Every measurable quantity, ~~including~~ position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by		The measurement of some property amounts to acting on this vector with an "operator". Every measurable quantity, like position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by

	<math>{\hat{A}\hat{B}-\hat{B}\hat{A}}</math>.		<math>{\hat{A}\hat{B}-\hat{B}\hat{A}}</math>.
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	===Simple===		===Simple===
			A particle is confined to a box 0.1 nm wide. Estimate the minimum uncertainty of its momentum.

			Solution: The minimum uncertainties are related by <math> {\Delta x \Delta p \geq \hbar/2} </math>. Taking the uncertainty of the particle's position to be the width of the box, the uncertainty of the momentum is then

			<math> {\Delta p \geq \frac{\hbar}{2\Delta x}\approx 5.27\times 10^-25} </mat> kg m/s.

	===Middling===		===Middling===
	===Difficult===		===Difficult===

	==Connectedness==		==Connectedness==
	#~~Heisenburg~~ uncertainty principle later plays into quantum tunneling, where this inherent uncertainty in a particle may sometimes allow it to overcome a potential energy barrier it otherwise would not be able to overcome classically.		#The Heisenberg uncertainty principle later plays into quantum tunneling, where this inherent uncertainty in a particle may sometimes allow it to overcome a potential energy barrier it otherwise would not be able to overcome classically (see [[Quantum Tunneling through Potential Barriers]]).
	#As a consequence of Heisenberg uncertainty principle, microelectronics are limited in size because electrons are able to tunnel through transistors that are small enough for uncertainty principle to have an effect.		#As a consequence of Heisenberg uncertainty principle, microelectronics are limited in size because electrons are able to tunnel through transistors that are small enough for uncertainty principle to have an effect.
			#Hydrogen nuclei are positively charged, so bringing two of them close enough together to fuse requires passing an [[Electric Potential]]. This takes a large amount of kinetic energy, corresponding to a temperature much higher than the cores of most stars. Tunneling allows particles with much lower kinetic energies to fuse. In other words, without tunneling, stars including our sun would produce no energy at all!
	==History==		==History==

	The development of the Heisenberg uncertainty principle is largely accredited to Werner Heisenberg.		The development of the Heisenberg uncertainty principle is largely accredited to Werner Heisenberg.e

	===Single-slit Diffraction Experiments===		===Single-slit Diffraction Experiments===

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pictures

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	A quantum system, which in this case can be a single particle, is represented by a "state vector". The components of this vector contain information on the probabilities associated with measurements. Each "dimension" of this vector corresponds to a possible outcome, and the magnitudes of the vector along each dimension indicates the likelihood the particle will be found in the corresponding state.		A quantum system, which in this case can be a single particle, is represented by a "state vector". The components of this vector contain information on the probabilities associated with measurements. Each "dimension" of this vector corresponds to a possible outcome, and the magnitudes of the vector along each dimension indicates the likelihood the particle will be found in the corresponding state.

			[[File:Colorvector.jpg\|border\|caption]]

			Above is a graph depicting the components of a quantum state vector. Each component corresponds to an outcome of a measurement, which in this case is the fictitious property "color". It is evident that it is more likely the particle will be found to be "blue" rather than "red" or "green". Below is a graph depicting the components of a state vector for a continuous property, like position. Every point on the curve corresponds to a component of the vector (it has an infinite number of dimensions). For example, the value at the x-value 3.32 gives us information on the likelihood the particle will be found around x=3.32.

			[[File:Positionvector.jpg\|border\|caption]]

			An actual quantum state vector actually takes complex values for its components. It would be more accurate to add another dimension to both of the above graphs, indicating the imaginary part of each component. This has been omitted for simplicity.

	The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by		The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by

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	[[File:Colorvector.png\|left\|An illustration of a state vector under the fictitious property "color". It is apparent that the outcome "blue" is more likely than the other two outcomes of measurement "red" or "green".]]

	The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by		The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by

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	[[File:Colorvector.png~~\|300px~~\|left\|An illustration of a state vector under the fictitious property "color". It is apparent that the outcome "blue" is more likely than the other two outcomes of measurement "red" or "green".]]		[[File:Colorvector.png\|left\|An illustration of a state vector under the fictitious property "color". It is apparent that the outcome "blue" is more likely than the other two outcomes of measurement "red" or "green".]]

	The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by		The measurement amounts to acting on this vector with an "operator". Every measurable quantity, including position, momentum, energy, and even angular momentum can be paired with a corresponding operator. What is important is the fact operators can alter the components of the vector, and thus the particle itself. It is not physically possible to measure two entirely distinct quantities at once, so measuring position and momentum means applying one operator and then another (or vice versa). It is then easy to see that the order of application matters. The way two operators can alter (or preserve) a particle's state is summarized by a mathematical object called the commutator. For two objects <math>{\hat{A}}</math> and <math>{\hat{B}}</math>, the commutator is given by