The Born Rule - Revision history

Rehaan Naik at 09:13, 29 November 2022

2022-11-29T09:13:34Z

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	Given a state <math> \ket{\psi} = \frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-} </math>, calculate the probability that a measurement of the state comes in the state <math> \ket{+} </math>.		Given a state <math> \ket{\psi} = \frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-} </math>, calculate the probability that a measurement of the state comes in the state <math> \ket{+} </math>.
	Now, from above, we can assume that since we have expressed <math> \ket{\psi} </math> in the basis <math> {\ket{+}, \ket{-}} </math>, they must be a complete basis of the Hilbert Space and are the eigenvectors of the observable. Regardless, the probability, as per Born's rule, is as follows:		Now, from above, we can assume that since we have expressed <math> \ket{\psi} </math> in the basis <math> {\ket{+}, \ket{-}} </math>, they must be a complete basis of the Hilbert Space and are the eigenvectors of the observable. Regardless, the probability, as per Born's rule, is as follows:
	<math> \|\bra{+}\ket{\psi}\|^2 = \|\bra{+}[\frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-}]\|^2 </math>		<math> \|\bra{+}\ket{\psi}\|^2 = \|\bra{+}[\frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-}]\|^2 </math> <br>
	A feature of quantum observables is that they are represented by Hermitian matrices, whose eigenvalues have orthonormal eigenvectors. Since t, the basis vectors are orthonormal to each other. <br>		A feature of quantum observables is that they are represented by Hermitian matrices, whose eigenvalues have orthonormal eigenvectors. Since the outcomes (eigenvalues) have the corresponding state vector as the basis vectors, the basis vectors are orthonormal to each other. <br>
	Remember that the inner product is a generalization of the dot product - so the inner product of orthogonal state vectors:		Remember that the inner product is a generalization of the dot product - so the inner product of orthogonal state vectors:
	<math> < \bra{+}\ket{-}>= \|< \bra{+}\ket{-}>\|^2 = 0 </math>		<math> < \bra{+}\ket{-}>= \|< \bra{+}\ket{-}>\|^2 = 0 </math>

Rehaan Naik at 09:12, 29 November 2022

2022-11-29T09:12:23Z

← Older revision		Revision as of 05:12, 29 November 2022
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	Given a state <math> \ket{\psi} = \frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-} </math>, calculate the probability that a measurement of the state comes in the state <math> \ket{+} </math>.		Given a state <math> \ket{\psi} = \frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-} </math>, calculate the probability that a measurement of the state comes in the state <math> \ket{+} </math>.
	Now, from above, we can assume that since we have expressed <math> \ket{\psi} </math> in the basis <math> {\ket{+}, \ket{-}} </math>, they must be a complete basis of the Hilbert Space and are the eigenvectors of the observable. Regardless, the probability, as per Born's rule, is as follows:		Now, from above, we can assume that since we have expressed <math> \ket{\psi} </math> in the basis <math> {\ket{+}, \ket{-}} </math>, they must be a complete basis of the Hilbert Space and are the eigenvectors of the observable. Regardless, the probability, as per Born's rule, is as follows:
	<math> \|\bra{+}}\ket{\psi}\|^2 = \|\bra{+}[\frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-}]\|^2 </math>		<math> \|\bra{+}\ket{\psi}\|^2 = \|\bra{+}[\frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-}]\|^2 </math>
	A feature of quantum observables is that they are represented by Hermitian matrices, whose eigenvalues have orthonormal eigenvectors. Since t, the basis vectors are orthonormal to each other. <br>		A feature of quantum observables is that they are represented by Hermitian matrices, whose eigenvalues have orthonormal eigenvectors. Since t, the basis vectors are orthonormal to each other. <br>
	Remember that the inner product is a generalization of the dot product - so the inner product of orthogonal state vectors:		Remember that the inner product is a generalization of the dot product - so the inner product of orthogonal state vectors:

Rehaan Naik at 09:11, 29 November 2022

2022-11-29T09:11:22Z

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	==Examples==		==Examples==
	''Simple'' <br>		===Simple=== <br>
			Given a state <math> \ket{\psi} = \frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-} </math>, calculate the probability that a measurement of the state comes in the state <math> \ket{+} </math>.
	'~~'Middling''~~ <br>		Now, from above, we can assume that since we have expressed <math> \ket{\psi} </math> in the basis <math> {\ket{+}, \ket{-}} </math>, they must be a complete basis of the Hilbert Space and are the eigenvectors of the observable. Regardless, the probability, as per Born's rule, is as follows:
	~~''Difficult''~~<br>		<math> \|\bra{+}}\ket{\psi}\|^2 = \|\bra{+}[\frac{1}{2}\ket{+} + \frac{i\sqrt{3}}{2}\ket{-}]\|^2 </math>
			A feature of quantum observables is that they are represented by Hermitian matrices, whose eigenvalues have orthonormal eigenvectors. Since t, the basis vectors are orthonormal to each other. <br>
			Remember that the inner product is a generalization of the dot product - so the inner product of orthogonal state vectors:
			<math> < \bra{+}\ket{-}>= \|< \bra{+}\ket{-}>\|^2 = 0 </math>
			And the inner product of a vector with itself:
			<math> < \bra{+}\ket{+}>= \|< \bra{+}\ket{+}>\|^2 = 1 </math>
			Thus, the probability <math> = \|\bra{+}\frac{1}{2}\ket{+} + \bra{+}\frac{i\sqrt{3}}{2}\ket{-}]\|^2 = \|\frac{1}{2}\bra{+}\ket{+}\|^2 = \|\frac{1}{2}\|^2 = \frac{1}{4} </math>
			===Middling===
			===Difficult===
	==Connectedness==		==Connectedness==
	How is this topic connected to something that you are interested in?		How is this topic connected to something that you are interested in?

Rehaan Naik: /* Born Rule as the Inner Product */

2022-11-28T03:56:15Z

Born Rule as the Inner Product

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	For an arbitrary state <math> \ket{\psi} </math>, we now have the Born rule: <br>		For an arbitrary state <math> \ket{\psi} </math>, we now have the Born rule: <br>
	<math> Prob_{\psi}(A = a) = \|\bra{\phi_a}\ket{\psi}\|^2 </math> <br>		<math> Prob_{\psi}(A = a) = \|\bra{\phi_a}\ket{\psi}\|^2 </math> <br>
			The expression on the right is the norm squared of the ''inner product''.
			===Inner Product===
			Note that both <math> \ket{\psi} </math> and <math> \ket{\phi_a} </math> are state ''vectors''. The inner product is a generalization of dot products, and is the multiplication of the conjugate transpose of the vector on the left and the vector on the right. <br>
			Conjugate transpose of <math> \ket{\psi} = \bra{\psi} </math> <br>
			A note: <br>

			<math> \|\bra{\psi}\ket{\psi}\|^2 = \bra{\psi}\ket{\psi} = 1 </math>

	==The Wavefunction as a Continuous Representation of State==		==The Wavefunction as a Continuous Representation of State==
	The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>		The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>

Rehaan Naik at 03:45, 28 November 2022

2022-11-28T03:45:36Z

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	This result is particularly important because it allows physicists to test their predictions about quantum systems, making quantum mechanics a testable theory.		This result is particularly important because it allows physicists to test their predictions about quantum systems, making quantum mechanics a testable theory.

	==Notation and Formalism==		==Notation and Formalism for Discrete Observables==
	Quantum systems, and all the values and possible states that they can hold, are associated with ''Hilbert Spaces''. These contain the states that a quantum system can have, and are vector spaces. Then, each quantum state can be represented as a vector in this Hilbert Space. (Because these are vectors in a vector space, linear combinations of quantum states also produce quantum states - within some constraints - we require that the norm (a generalized idea of length) be 1, making the state vector essentially a unit vector). <br>		Quantum systems, and all the values and possible states that they can hold, are associated with ''Hilbert Spaces''. These contain the states that a quantum system can have, and are vector spaces. Then, each quantum state can be represented as a vector in this Hilbert Space. (Because these are vectors in a vector space, linear combinations of quantum states also produce quantum states - within some constraints - we require that the norm (a generalized idea of length) be 1, making the state vector essentially a unit vector). <br>
	Then a state <math> \psi </math> is represented by the state vector <math> \ket{\psi} </math>		Then a state <math> \psi </math> is represented by the state vector <math> \ket{\psi} </math> <br>
	Now let's consider observables, such as position or momentum or angular momentum. Let us represent such an observable as <math> A </math>.		Now let's consider observables, such as position or momentum or angular momentum. Let us represent such an observable as <math> A </math>. Then for a discrete observable, there are a discrete set of outcomes that we represent as <math> {a} </math>. For discrete observables, <math> A </math> is a matrix, and we have a set of vectors <math> {\ket{\phi_a}} </math> that are eigenvectors of <math> A </math> corresponding to the eigenvalues <math> {a} </math>. <br>
			<math> A{\ket{\phi_a}} = a\ket{\phi_a} </math> <br>
			Then we have a neat little realization: <br>
			<math> Prob_{\phi_a}(A = a) = 1 </math> <br>
			The expression above refers to the probability of the state <math> \ket{\phi_a} </math> having the outcome <math> a </math> for the observable <math> A </math>
	==Born Rule as the Inner Product==		==Born Rule as the Inner Product==
			For an arbitrary state <math> \ket{\psi} </math>, we now have the Born rule: <br>
			<math> Prob_{\psi}(A = a) = \|\bra{\phi_a}\ket{\psi}\|^2 </math> <br>
	==The Wavefunction as a Continuous Representation of State==		==The Wavefunction as a Continuous Representation of State==
	The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>		The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>

Rehaan Naik at 03:31, 28 November 2022

2022-11-28T03:31:22Z

← Older revision		Revision as of 23:31, 27 November 2022
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	==Notation and Formalism==		==Notation and Formalism==
	Quantum systems, and all the values and possible states that they can hold, are associated with ''Hilbert Spaces''. These contain the states that a quantum system can have, and are vector spaces. Then, each quantum state can be represented as a vector in this Hilbert Space.		Quantum systems, and all the values and possible states that they can hold, are associated with ''Hilbert Spaces''. These contain the states that a quantum system can have, and are vector spaces. Then, each quantum state can be represented as a vector in this Hilbert Space. (Because these are vectors in a vector space, linear combinations of quantum states also produce quantum states - within some constraints - we require that the norm (a generalized idea of length) be 1, making the state vector essentially a unit vector). <br>
			Then a state <math> \psi </math> is represented by the state vector <math> \ket{\psi} </math>
			Now let's consider observables, such as position or momentum or angular momentum. Let us represent such an observable as <math> A </math>.
	==Born Rule as the Inner Product==		==Born Rule as the Inner Product==
	==The Wavefunction as a Continuous Representation of State==		==The Wavefunction as a Continuous Representation of State==

Rehaan Naik at 03:18, 28 November 2022

2022-11-28T03:18:50Z

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	'''Claimed - Rehaan Naik 11/27/2022'''		'''Claimed - Rehaan Naik 11/27/2022''' <br>
	The Born rule provides a link between the mathematical formalism of quantum theory and experiment, and as such is almost single-handedly responsible for practically all predictions of quantum physics. <br>		The Born rule provides a link between the mathematical formalism of quantum theory and experiment, and as such is almost single-handedly responsible for practically all predictions of quantum physics. <br>
	It is an important result of quantum mechanics that describes the probability density of a measured quantum system. In particular, it states that the square of the wavefunction is proportional to the probability density function. This result is a continuous form of the norm squared of the inner product being the probability that a known state takes the value of another. <br>		It is an important result of quantum mechanics that describes the probability density of a measured quantum system. In particular, it states that the square of the wavefunction is proportional to the probability density function. This result is a continuous form of the norm squared of the inner product being the probability that a known state takes the value of another. <br>
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	This result is particularly important because it allows physicists to test their predictions about quantum systems, making quantum mechanics a testable theory.		This result is particularly important because it allows physicists to test their predictions about quantum systems, making quantum mechanics a testable theory.

			==Notation and Formalism==
			Quantum systems, and all the values and possible states that they can hold, are associated with ''Hilbert Spaces''. These contain the states that a quantum system can have, and are vector spaces. Then, each quantum state can be represented as a vector in this Hilbert Space.
			==Born Rule as the Inner Product==
	==The Wavefunction as a Continuous Representation of State==		==The Wavefunction as a Continuous Representation of State==
	The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>		The wavefunction is a complex function. However, there is no physical interpretation of complex numbers. Converting a complex number to a real number involves taking the norm of the complex number. Now, see that the wavefunction <math> \psi(x) </math> is the continuous analogue of <math> \ket{\psi} </math> in the x basis. <math> \ket{\psi} </math> is a state vector for which the norm is finite, and the inner product is defined within the Hilbert Space. We know that the norm squared of the inner product of a state vector with itself is 1 since we have defined state vectors to have norm 1. Since the wavefunction is the analogue of the state vector, it makes sense that an analogue of the inner product is defined for it - the inner product of a state vector with itself is the multiplication of the the conjugate transpose of the state vector with itself. So, the norm squared of the wavefunction comes from the multiplication of the wavefunction with its complex conjugate. <br>

Rehaan Naik: /* References */

2022-11-28T03:07:10Z

References

← Older revision		Revision as of 23:07, 27 November 2022
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	Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [https://www.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>		Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [https://www.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>
	Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>		Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>
	Townsend, J S. ~~Amodern~~ Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.		Townsend, J S. A Modern Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.

Rehaan Naik: /* References */

2022-11-28T03:06:48Z

References

← Older revision		Revision as of 23:06, 27 November 2022
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	==See also==		==See also==
	==References==		==References==
	Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [https://.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>		Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [https://www.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>
	Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>		Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>
	Townsend, J S. Amodern Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.		Townsend, J S. Amodern Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.

Rehaan Naik: /* References */

2022-11-28T03:06:23Z

References

← Older revision		Revision as of 23:06, 27 November 2022
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	==See also==		==See also==
	==References==		==References==
	~~Works Cited~~ Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [~~www~~.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>		Ball, Phillip. “The Born Rule Has Been Derived from Simple Physical Principles.” Quanta Magazine, 13 Feb. 2019, [https://.quantamagazine.org/the-born-rule-has-been-derived-from-simple-physical-principles-20190213/] <br>
	Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>		Landsman, N.P. The Born Rule and Its Interpretation. [https://www.math.ru.nl/~landsman/Born.pdf] <br>
	Townsend, J S. Amodern Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.		Townsend, J S. Amodern Approach to Quantum Mechancs. Saisalito, Ca, University Science Books, 2000.