Quantum Mechanics For Engineers - Baylor ECS

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OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Quantum Mechanics for EngineersMatt RobinsonBaylor UniversityDepartment of PhysicsMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Outline1What is Classical Physics?Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Outline1What is Classical Physics?2What is Quantum Physics?Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Outline1What is Classical Physics?2What is Quantum Physics?3How Can This Apply to Computers?Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsActions and The Principle of Least ActionKnowing the (scalar) expressions for the kinetic energy T (q̇)and the potential energy V (q) for a system, we can define theLagrangian asL(q, q̇) T VMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsActions and The Principle of Least ActionKnowing the (scalar) expressions for the kinetic energy T (q̇)and the potential energy V (q) for a system, we can define theLagrangian asL(q, q̇) T VThe Action S[q] is then defined as the integral over time,ZS[q] dt L(q, q̇)Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsActions and The Principle of Least ActionThe equations of motion are then given by the zeros of theEuler Lagrange Derivative,d L LδL 0δqdt q̇ qMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsActions and The Principle of Least ActionThe equations of motion are then given by the zeros of theEuler Lagrange Derivative,d L LδL 0δqdt q̇ qThis is called the principle of Least Action.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsProjectile MotionT 12 m(ẋ 2 ẏ 2 ),V mgyMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsProjectile MotionT 12 m(ẋ 2 ẏ 2 ),L T V 122 m(ẋV mgy ẏ 2 ) mgyMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsProjectile MotionT 12 m(ẋ 2 ẏ 2 ),L T V 122 m(ẋd Ldt ẋ mẍ, ddt mẋV mgy ẏ 2 ) mgy L x 0 mẍ 0Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsProjectile MotionT 12 m(ẋ 2 ẏ 2 ),V mgyL T V 122 m(ẋd Ldt ẋ mẍ, L x mÿ , L y ddt mẋ ddt mẏ ẏ 2 ) mgy 0 mẍ 0d Ldt ẏ mg mÿ mgMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsConcluding Thoughts on Classical PhysicsMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsConcluding Thoughts on Classical PhysicsFor a given initial condition,only one path is possible - thepath which satisfies theEuler-Lagrange Equation.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsConcluding Thoughts on Classical PhysicsFor a given initial condition,only one path is possible - thepath which satisfies theEuler-Lagrange Equation.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Actions and The Principle of Least ActionExampleConcluding Thoughts on Classical PhysicsConcluding Thoughts on Classical PhysicsFor a given initial condition,only one path is possible - thepath which satisfies theEuler-Lagrange Equation.The nature of classical physicscan be thought of as follows:INPUT: Ask question.OUTPUT: Get answer.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsSo What is Quantum Physics?Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyClassical physics demandedδL 0δqMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyClassical physics demandedδL 0δqMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyClassical physics demandedδL 0δqQuantum physics allows anyvalue on the right hand sideδL λδqwhere λ can be anything.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyClassical physics demandedδL 0δqQuantum physics allows anyvalue on the right hand sideδL λδqwhere λ can be anything.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyClassical physics demandedδL 0δqQuantum physics allows anyvalue on the right hand sideδL λδqwhere λ can be anything.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyHowever, not all paths are equally probable.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsPrinciple of Least Action . . . ProbablyHowever, not all paths are equally probable.Each possible path has a statistical weight/probability equal toe S[q]/ ore iS[q]/ (these are the same under the change of variables dt idt)Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsBut it gets worseMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsInterferenceq e S[q]/ Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsInterferenceqq δq Matt Robinsone S[q]/ e S[q δq]/ Quantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsInterferenceqq δqe S[q δq]/ ee S[q]/ e S[q δq]/ (S[q] δqδS[q])/ δqMatt Robinson e S[q]/ e δq δS[q]δqQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsInterferenceqq δqe S[q δq]/ ee S[q]/ e S[q δq]/ (S[q] δqδS[q])/ δq e S[q]/ e δq δS[q]δqMost likely path is whereδS[q] 0δq(which is the same as whereMatt RobinsonδLδq 0)Quantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsEquation of StateIf ψ e iS/ , then ψi S ψ t tMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsEquation of StateIf ψ e iS/ , then ψi S ψ t tHψ i ψ t(where H T V )Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsEquation of StateIf ψ e iS/ , then ψi S ψ t tHψ i ψ t(where H T V )Solving this differential equation (called Schroedinger’s Equation)for ψ will give the (complex) Wave Function for the particleMatt RobinsonQuantum Mechanics for Engineers

Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsOutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Equation of StateIf ψ e iS/ , then ψi S ψ t tHψ i ψ t(where H T V )Solving this differential equation (called Schroedinger’s Equation)for ψ will give the (complex) Wave Function for the particleZb?Zψ ψ dx ab ψ 2 dxaMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Matt RobinsonPrinciple of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Matt RobinsonPrinciple of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Matt RobinsonPrinciple of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Matt RobinsonPrinciple of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsHydrogen AtomMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsHydrogen AtomMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsObservationBut it gets even worse .Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsObservationA system exists in every possible state it can be in prior toobservation (superposition).Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsObservationA system exists in every possible state it can be in prior toobservation (superposition).These possibilities all interfere with each other.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsObservationA system exists in every possible state it can be in prior toobservation (superposition).These possibilities all interfere with each other.Once an observation is made, the system “collapses” into oneof the possible states, according to some probabilitydistribution which depends on the system.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsConcluding Thoughts on Quantum PhysicsFor a given initial condition, any path/state is possible - nomatter how improbable it may be.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsConcluding Thoughts on Quantum PhysicsFor a given initial condition, any path/state is possible - nomatter how improbable it may be.How often you will observe that path/state is determined bythe wave function, which is a probability distribution.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsConcluding Thoughts on Quantum PhysicsFor a given initial condition, any path/state is possible - nomatter how improbable it may be.How often you will observe that path/state is determined bythe wave function, which is a probability distribution.The classical path/state will be the most likely, which is whymacroscopic systems appear classical.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Principle of Least Action . . . ProbablySuperpositionSo How Do We Do Quantum Mechanics?ObservationConcluding Thoughts on Quantum PhysicsConcluding Thoughts on Quantum PhysicsFor a given initial condition, any path/state is possible - nomatter how improbable it may be.How often you will observe that path/state is determined bythe wave function, which is a probability distribution.The classical path/state will be the most likely, which is whymacroscopic systems appear classical.The nature of quantum physics can be thought of as follows:INPUT: Ask question, suggest an answer.OUTPUT: How often that answer will be right.Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Classical Bits and Quantum BitsClassical computer works using bits, which can either be 0 or 1Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Classical Bits and Quantum BitsClassical computer works using bits, which can either be 0 or 1A quantum computer uses quantum bits, which can be 0 and1 ψi α 0i β 1iMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Classical Bits and Quantum BitsClassical computer works using bits, which can either be 0 or 1A quantum computer uses quantum bits, which can be 0 and1 ψi α 0i β 1iMeasurement produces:Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Classical Bits and Quantum BitsClassical computer works using bits, which can either be 0 or 1A quantum computer uses quantum bits, which can be 0 and1 ψi α 0i β 1iMeasurement produces:0 with probability α 2Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Classical Bits and Quantum BitsClassical computer works using bits, which can either be 0 or 1A quantum computer uses quantum bits, which can be 0 and1 ψi α 0i β 1iMeasurement produces:0 with probability α 21 with probability β 2Matt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Manipulating Q-BitsMatt RobinsonQuantum Mechanics for Engineers

OutlineWhat is Classical Physics?What is Quantum Physics?How Can This Apply to Computers?Manipulating Q-BitsAny (invertible) transformation done to ψi will act on everyterm in the superposition:Matt RobinsonQuantum Mechanic

What is Quantum Physics? How Can This Apply to Computers? Principle of Least Action .Probably Superposition So How Do We Do Quantum Mechanics? Observation Concluding Thoughts on Quantum Physics Principle of Least Action .Probably Classical physics demanded L q 0 Quantum physics allows any value on the right hand side L q where can be .

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