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Lecture 5 Motion of a Charged Particle in a Magnetic Field

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Lecture 5 Motion of a Charged Particle in a Magnetic Field Lecture 5 Motion of a charged particle in a magnetic field Charged particle in a magnetic field: Outline 1 Canonical quantization: lessons from classical dynamics 2 Quantum mechanics of a particle in a field 3 Atomic hydrogen in a uniform field: Normal Zeeman effect 4 Gauge invariance and the Aharonov-Bohm effect 5 Free electrons in a magnetic field: Landau levels 6 Integer Quantum Hall effect Lorentz force What is effect of a static electromagnetic field on a charged particle? Classically, in electric and magnetic field, particles experience a Lorentz force: F = q (E + v B) × q denotes charge (notation: q = e for electron). − Velocity-dependent force qv B very different from that derived from scalar potential, and programme× for transferring from classical to quantum mechanics has to be carried out with more care. As preparation, helpful to revise(?) how the Lorentz force law arises classically from Lagrangian formulation. Analytical dynamics: a short primer For a system with m degrees of freedom specified by coordinates q , q , classical action determined from Lagrangian L(q , q˙ ) by 1 ··· m i i S[qi ]= dt L(qi , q˙ i ) ! For conservative forces (those which conserve mechanical energy), L = T V , with T the kinetic and V the potential energy. − Hamilton’s extremal principle: trajectories qi (t) that minimize action specify classical (Euler-Lagrange) equations of motion, d (∂ L(q , q˙ )) ∂ L(q , q˙ )=0 dt q˙ i i i − qi i i mq˙ 2 e.g. for a particle in a potential V (q), L(q, q˙ )= V (q) and 2 − from Euler-Lagrange equations, mq¨ = ∂ V (q) − q Analytical dynamics: a short primer For a system with m degrees of freedom specified by coordinates q , q , classical action determined from Lagrangian L(q , q˙ ) by 1 ··· m i i S[qi ]= dt L(qi , q˙ i ) ! For conservative forces (those which conserve mechanical energy), L = T V , with T the kinetic and V the potential energy. − Hamilton’s extremal principle: trajectories qi (t) that minimize action specify classical (Euler-Lagrange) equations of motion, d (∂ L(q , q˙ )) ∂ L(q , q˙ )=0 dt q˙ i i i − qi i i mq˙ 2 e.g. for a particle in a potential V (q), L(q, q˙ )= V (q) and 2 − from Euler-Lagrange equations, mq¨ = ∂ V (q) − q Analytical dynamics: a short primer To determine the classical Hamiltonian H from the Lagrangian, first obtain the canonical momentum pi = ∂q˙ i L and then set H(q , p )= p q˙ L(q , q˙ ) i i i i − i i "i mq˙ 2 e.g. for L(q, q˙ )= 2 V (q), p = ∂q˙ L = mq˙ , and − 2 2 H = pq˙ L = p p ( p V (q)) = p + V (q). − m − 2m − 2m In Hamiltonian formulation, minimization of classical action S = dt pi q˙ i H(qi , pi ) , leads to Hamilton’s equations: # − $ ! "i q˙ = ∂ H, p˙ = ∂ H i pi i − qi i.e. if Hamiltonian is independent of qi , corresponding momentum pi is conserved, i.e. pi is a constant of the motion. Analytical dynamics: a short primer To determine the classical Hamiltonian H from the Lagrangian, first obtain the canonical momentum pi = ∂q˙ i L and then set H(q , p )= p q˙ L(q , q˙ ) i i i i − i i "i mq˙ 2 e.g. for L(q, q˙ )= 2 V (q), p = ∂q˙ L = mq˙ , and − 2 2 H = pq˙ L = p p ( p V (q)) = p + V (q). − m − 2m − 2m In Hamiltonian formulation, minimization of classical action S = dt pi q˙ i H(qi , pi ) , leads to Hamilton’s equations: # − $ ! "i q˙ = ∂ H, p˙ = ∂ H i pi i − qi i.e. if Hamiltonian is independent of qi , corresponding momentum pi is conserved, i.e. pi is a constant of the motion. Analytical dynamics: a short primer To determine the classical Hamiltonian H from the Lagrangian, first obtain the canonical momentum pi = ∂q˙ i L and then set H(q , p )= p q˙ L(q , q˙ ) i i i i − i i "i mq˙ 2 e.g. for L(q, q˙ )= 2 V (q), p = ∂q˙ L = mq˙ , and − 2 2 H = pq˙ L = p p ( p V (q)) = p + V (q). − m − 2m − 2m In Hamiltonian formulation, minimization of classical action S = dt pi q˙ i H(qi , pi ) , leads to Hamilton’s equations: # − $ ! "i q˙ = ∂ H, p˙ = ∂ H i pi i − qi i.e. if Hamiltonian is independent of qi , corresponding momentum pi is conserved, i.e. pi is a constant of the motion. Analytical dynamics: a short primer To determine the classical Hamiltonian H from the Lagrangian, first obtain the canonical momentum pi = ∂q˙ i L and then set H(q , p )= p q˙ L(q , q˙ ) i i i i − i i "i mq˙ 2 e.g. for L(q, q˙ )= 2 V (q), p = ∂q˙ L = mq˙ , and − 2 2 H = pq˙ L = p p ( p V (q)) = p + V (q). − m − 2m − 2m In Hamiltonian formulation, minimization of classical action S = dt pi q˙ i H(qi , pi ) , leads to Hamilton’s equations: # − $ ! "i q˙ = ∂ H, p˙ = ∂ H i pi i − qi i.e. if Hamiltonian is independent of qi , corresponding momentum pi is conserved, i.e. pi is a constant of the motion. Analytical dynamics: Lorentz force As Lorentz force F = qv B is velocity dependent, it can not be expressed as gradient of some× potential – nevertheless, classical equations of motion still specifed by principle of least action. With electric and magnetic fields written in terms of scalar and vector potential, B = A, E = ϕ ∂ A, Lagrangian: ∇× −∇ − t 1 L = mv2 qϕ + qv A 2 − · q x =(x , x , x ) andq ˙ v = (x ˙ , x˙ , x˙ ) i ≡ i 1 2 3 i ≡ i 1 2 3 N.B. form of Lagrangian more natural in relativistic formulation: qv µA = qϕ + qv A where v µ =(c, v) and Aµ =(ϕ/c, A) − µ − · Canonical momentum: pi = ∂x˙i L = mvi + qAi no longer given by mass velocity – there is an extra term! × Analytical dynamics: Lorentz force As Lorentz force F = qv B is velocity dependent, it can not be expressed as gradient of some× potential – nevertheless, classical equations of motion still specifed by principle of least action. With electric and magnetic fields written in terms of scalar and vector potential, B = A, E = ϕ ∂ A, Lagrangian: ∇× −∇ − t 1 L = mv2 qϕ + qv A 2 − · q x =(x , x , x ) andq ˙ v = (x ˙ , x˙ , x˙ ) i ≡ i 1 2 3 i ≡ i 1 2 3 N.B. form of Lagrangian more natural in relativistic formulation: qv µA = qϕ + qv A where v µ =(c, v) and Aµ =(ϕ/c, A) − µ − · Canonical momentum: pi = ∂x˙i L = mvi + qAi no longer given by mass velocity – there is an extra term! × Analytical dynamics: Lorentz force As Lorentz force F = qv B is velocity dependent, it can not be expressed as gradient of some× potential – nevertheless, classical equations of motion still specifed by principle of least action. With electric and magnetic fields written in terms of scalar and vector potential, B = A, E = ϕ ∂ A, Lagrangian: ∇× −∇ − t 1 L = mv2 qϕ + qv A 2 − · q x =(x , x , x ) andq ˙ v = (x ˙ , x˙ , x˙ ) i ≡ i 1 2 3 i ≡ i 1 2 3 N.B. form of Lagrangian more natural in relativistic formulation: qv µA = qϕ + qv A where v µ =(c, v) and Aµ =(ϕ/c, A) − µ − · Canonical momentum: pi = ∂x˙i L = mvi + qAi no longer given by mass velocity – there is an extra term! × Analytical dynamics: Lorentz force As Lorentz force F = qv B is velocity dependent, it can not be expressed as gradient of some× potential – nevertheless, classical equations of motion still specifed by principle of least action. With electric and magnetic fields written in terms of scalar and vector potential, B = A, E = ϕ ∂ A, Lagrangian: ∇× −∇ − t 1 L = mv2 qϕ + qv A 2 − · q x =(x , x , x ) andq ˙ v = (x ˙ , x˙ , x˙ ) i ≡ i 1 2 3 i ≡ i 1 2 3 N.B. form of Lagrangian more natural in relativistic formulation: qv µA = qϕ + qv A where v µ =(c, v) and Aµ =(ϕ/c, A) − µ − · Canonical momentum: pi = ∂x˙i L = mvi + qAi no longer given by mass velocity – there is an extra term! × Analytical dynamics: Lorentz force From H(q , p )= p q˙ L(q , q˙ ), Hamiltonian given by: i i i i − i i "i 1 1 H = (mv + qA ) v mv2 qϕ + qv A = mv2 + qϕ i i i − 2 − · 2 i ) * " = pi = L(˙qi , qi ) % &' ( % &' ( To determine classical equations of motion, H must be expressed solely in terms of coordinates and canonical momenta, p = mv + qA 1 H = (p qA(x, t))2 + qϕ(x, t) 2m − Then, from classical equations of motionx ˙i = ∂pi H and p˙ = ∂ H, and a little algebra, we recover Lorentz force law i − xi mx¨ = F = q (E + v B) × Analytical dynamics: Lorentz force From H(q , p )= p q˙ L(q , q˙ ), Hamiltonian given by: i i i i − i i "i 1 1 H = (mv + qA ) v mv2 qϕ + qv A = mv2 + qϕ i i i − 2 − · 2 i ) * " = pi = L(˙qi , qi ) % &' ( % &' ( To determine classical equations of motion, H must be expressed solely in terms of coordinates and canonical momenta, p = mv + qA 1 H = (p qA(x, t))2 + qϕ(x, t) 2m − Then, from classical equations of motionx ˙i = ∂pi H and p˙ = ∂ H, and a little algebra, we recover Lorentz force law i − xi mx¨ = F = q (E + v B) × Analytical dynamics: Lorentz force From H(q , p )= p q˙ L(q , q˙ ), Hamiltonian given by: i i i i − i i "i 1 1 H = (mv + qA ) v mv2 qϕ + qv A = mv2 + qϕ i i i − 2 − · 2 i ) * " = pi = L(˙qi , qi ) % &' ( % &' ( To determine classical equations of motion, H must be expressed solely in terms of coordinates and canonical momenta, p = mv + qA 1 H = (p qA(x, t))2 + qϕ(x, t) 2m − Then, from classical equations of motionx ˙i = ∂pi H and p˙ = ∂ H, and a little algebra, we recover Lorentz force law i − xi mx¨ = F = q (E + v B) × Lessons from classical dynamics So, in summary, the classical Hamiltonian for a charged particle in an electromagnetic field is given by 1 H = (p qA(x, t))2 + qϕ(x, t) 2m − This is all that you need to recall – its first principles derivation from the Lagrangian formulation is not formally examinable! Using this result as a platform, we can now turn to the quantum mechanical formulation.
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