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Physics 3 for Electrical Engineering Ben Gurion University of the Negev www.bgu.ac.il/atomchipwww.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenterwww.bgu.ac.il/nanocenter Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin Week 9. Quantum mechanics – angular momentum operators commutation relations eigenvalues and eigenvectors of Sources: Merzbacher (2 nd edition) Chap. 9; Merzbacher (3 rd edition) Chap. 11.
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The time-independent Schrödinger equation for a particle in three dimensions is where r = (x,y,z), and A special, but important, class of potentials is the class of central potentials, which depend only on r = (x 2 + y 2 + z 2 ) 1/2, i.e.
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The time-independent Schrödinger equation for a particle in three dimensions is where r = (x,y,z), and A special, but important, class of potentials is the class of central potentials, which depend only on r = (x 2 + y 2 + z 2 ) 1/2, i.e.
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The time-independent Schrödinger equation for a particle in three dimensions is where r = (x,y,z), and A special, but important, class of potentials is the class of central potentials, which depend only on r = (x 2 + y 2 + z 2 ) 1/2, i.e.
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If the only force acting is a central force, we know from classical physics about an important conserved quantity: angular momentum. Angular momentum L = r × p is conserved because F(r) and r are always parallel:
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If the only force acting is a central force, we know from classical physics about an important conserved quantity: angular momentum. Angular momentum L = r × p is conserved because F(r) and r are always parallel: ? © 2005-2009 George CoghillGeorge Coghill
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If the only force acting is a central force, we know from classical physics about an important conserved quantity: angular momentum. Angular momentum L = r × p is conserved because F(r) and r are always parallel:
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If the only force acting is a central force, we know from classical physics about an important conserved quantity: angular momentum. Angular momentum L = r × p is conserved because F(r) and r are always parallel: Is there a quantum operator for angular momentum? Is it conserved?
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Angular momentum operators In classical mechanics, angular momentum is defined as L = r × p. So in quantum mechanics, is angular momentum defined as
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Angular momentum operators In classical mechanics, angular momentum is defined as L = r × p. So in quantum mechanics, is angular momentum defined as (Isn ’ t there a problem with the ordering of and ?) © 2005-2009 George CoghillGeorge Coghill ?
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Angular momentum operators In classical mechanics, angular momentum is defined as L = r × p. So in quantum mechanics, is angular momentum defined as
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Angular momentum operators In classical mechanics, angular momentum is defined as L = r × p. So in quantum mechanics, angular momentum is defined as
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Angular momentum operators In classical mechanics, angular momentum is defined as L = r × p. So in quantum mechanics, angular momentum is defined as Is conserved?
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Theorem: If a Hermitian operator commutes with, i.e., then we can find eigenstates of that are also eigenstates of. Proof: If commutes with an observable, then for any state,. So if is an eigenvector of with eigenvalue E n, i.e., then so is : If E n is nondegenerate, then for some number a, so is an eigenstate of. If E n is degenerate, then eigenstates of form a basis for the subspace of eigenvectors of with eigenvalue E n.
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Corollary: If a Hermitian operator commutes with, i.e., then the expectation value of in any state is constant in time, i.e. Proof: Define the state such that and Write and then does not depend on time.
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So is conserved? That is, do and commute? Let ’ s see:
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So is conserved? That is, do and commute? Let ’ s see: Remember: the derivatives act also on a wave function ψ. [ ψ] ψ ψ
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So is conserved? That is, do and commute? Let ’ s see:
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So is conserved? That is, do and commute? Let ’ s see:
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So is conserved? That is, do and commute? Let ’ s see:
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So is conserved? That is, do and commute? Let ’ s see: [ ψ] Remember: the derivatives act also on a wave function ψ. ψ ψ
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So is conserved? That is, do and commute? Let ’ s see: [ ψ] Remember: the derivatives act also on a wave function ψ. ψ ψ
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So is conserved? That is, do and commute? Let ’ s see:
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So is conserved? That is, do and commute? Let ’ s see: And what holds for holds for and.
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So is conserved! What are the possible values of ? Let ’ s calculate commutation relations for.
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Commutation relations using
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Commutation relations We can choose a basis of eigenstates of, or of, or of, but only one of these bases at a time! Also, from our generalized uncertainty principle, we conclude
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Eigenvalues and eigenvectors of. Let ’ s prove that is a raising operator: Similarly, is a lowering operator: Suppose. Then
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Eigenvalues and eigenvectors of. Let ’ s prove that is a raising operator: Similarly, is a lowering operator: Suppose. Then
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Eigenvalues and eigenvectors of. Let ’ s prove that is a raising operator: Similarly, is a lowering operator: Suppose. Then so is an eigenvector of with eigenvalue. Similarly, is an eigenvector of with eigenvalue.
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Eigenvalues and eigenvectors of. Eigenvalues of : but since L z is bounded by we must have for some m max and m min. Note since Note also
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Eigenvalues and eigenvectors of. To make the rest of the calculations easier, we should change to spherical coordinates: Now so The eigenfunctions of are with eigenvalues so Therefore m max = –m min and, by convention, m max = l.
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Eigenvalues and eigenvectors of : Summary: For a given value of l, the eigenstates of are with respective eigenvalues These 2l+1 eigenvectors of are also eigenvectors of with degenerate eigenvalue
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Schrödinger’s equation for a central potential is A vector identity: What is ? Since it is hence
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Schrödinger’s equation for a central potential is A vector identity: What is ? Since it is hence
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Solving for we obtain and by comparing this with the expression for in spherical coordinates,
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Solving for we obtain and by comparing this with the expression for in spherical coordinates,
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Solving for we obtain and by comparing this with the expression for in spherical coordinates we conclude
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Now back to Schrödinger’s equation in spherical coordinates: since the eigenvalues of are, we can solve this equation by expressing ψ(r,θ,φ) as a product of two functions: ψ(r,θ,φ) = R(r) Y l m (θ,φ), where
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Here are the lowest eigenfunctions Y l m (θ,φ) of
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HereHere’s how they look:
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