The Postulates. What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief. a theory postulated by a respected scientist synonyms: suggest, advance, posit, hypothesize, propose, assume 2 (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority. Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief. a theory postulated by a respected scientist synonyms: suggest, advance, posit, hypothesize, propose, assume 2 (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority. How do you know it s correct? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief. a theory postulated by a respected scientist synonyms: suggest, advance, posit, hypothesize, propose, assume 2 (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority. How do you know it s correct? DATA! Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief. a theory postulated by a respected scientist synonyms: suggest, advance, posit, hypothesize, propose, assume 2 (in ecclesiastical law) nominate or elect (someone) to an ecclesiastical office subject to the sanction of a higher authority. How do you know it s correct? DATA! See here for an example of impeccable logic. Jerry Gilfoyle The Rules of the Quantum Game 1 / 21

The Postulates (the Rules of the Game) 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. Jerry Gilfoyle The Rules of the Quantum Game 2 / 21

The Postulates (the Rules of the Game) 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. Jerry Gilfoyle The Rules of the Quantum Game 2 / 21

The Postulates (the Rules of the Game) 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. 3 The state of a system is represented by a wave function Ψ that is continuous, differentiable and contains all possible information about the system. The average value of any observable A is A = all space Ψ Â Ψd r. The intensity is proportional to Ψ 2. Jerry Gilfoyle The Rules of the Quantum Game 2 / 21

The Postulates (the Rules of the Game) 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. 3 The state of a system is represented by a wave function Ψ that is continuous, differentiable and contains all possible information about the system. The average value of any observable A is A = all space Ψ Â Ψd r. The intensity is proportional to Ψ 2. 4 The time development of the wave function is determined by Ψ( r, t) i = 2 t 2µ 2 Ψ( r, t) + V ( r)ψ( r, t) µ reduced mass. Jerry Gilfoyle The Rules of the Quantum Game 2 / 21

Interpreting the Quantum Intensity Electron diffraction by gold The square of the magnitude of the wave function ψ(x, t) 2 = ψ (x, t)ψ(x, t) is the probability density. Jerry Gilfoyle The Rules of the Quantum Game 3 / 21

Apply the Rules: A Particle in a Box Consider the infinite rectangular well potential shown in the figure below. What is the time-independent Schroedinger equation for this potential? What is the general solution to the previous question? V(x) What are the boundary conditions the solution must satisfy What is the particular solution for this potential? 0 a x What is the energy of the particular solution? Jerry Gilfoyle The Rules of the Quantum Game 4 / 21

The Infinite Rectangular Well Potential - Energy Levels Energy Levels n 10 9 8 Energy 7 6 5 4 3 2 1 Jerry Gilfoyle The Rules of the Quantum Game 5 / 21

Some Math The solutions of the Schroedinger equation form a Hilbert space. 1 They are linear, i.e. superposition/interference is built in. If a is a constant and φ(x) is an element of the space, then so is aφ(x). If φ 1 (x) and φ 2 (x) are elements, then so is φ 1 (x) + φ 2 (x). 2 An inner product is defined and all elements have a norm. φ n ψ = φ nψ dx and φ n φ n = 3 The solutions are complete. ψ = b n φ n n=0 4 The solutions are orthonormal so φ n φ n = δ n,n. The operators are Hermitian - their eigenvalues are real. φ nφ n dx = 1 Jerry Gilfoyle The Rules of the Quantum Game 6 / 21

Apply the Rules More: A Particle in a Box Consider the infinite rectangular well potential shown in the figure below with an initial wave packet defined in the following way. Ψ(x, 0) = 1 d x 0 < x < x 1 and d = x 1 x 0 = 0 otherwise What possible values are obtained in an energy measurement? V(x) Ψ(x) What eigenfunctions contribute to this wave packet and what are their probabilities? What will many measurements of the energy give? 0 x0 x1 a x Jerry Gilfoyle The Rules of the Quantum Game 7 / 21

The Infinite Rectangular Well Potential - Energy Levels Energy Levels n 10 9 8 Energy 7 6 5 4 3 2 1 Jerry Gilfoyle The Rules of the Quantum Game 8 / 21

L Hôpital s Rule If lim f (x) = lim g(x) = 0 or lim f (x) = lim g(x) = ± x c x c x c x c and exists, then f (x) lim x c g (x) f (x) lim x c g(x) = lim f (x) x c g (x). Jerry Gilfoyle The Rules of the Quantum Game 9 / 21

Truncated Fourier Series Probability 12 10 8 6 4 2 1 term 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 10 / 21

Truncated Fourier Series Probability 12 10 8 6 4 2 1 term 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Probability 12 10 8 6 4 2 5 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 10 / 21

Truncated Fourier Series Probability 12 10 8 6 4 2 1 term 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Probability 12 10 8 6 4 2 5 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Probability 12 10 8 6 4 2 25 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 10 / 21

Truncated Fourier Series Probability Probability 12 10 8 6 4 2 1 term 0 0.0 0.2 0.4 0.6 0.8 1.0 12 10 8 6 4 2 Position 25 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Probability Probability 12 10 8 6 4 2 5 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 12 10 8 6 4 2 Position 125 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 10 / 21

Truncated Fourier Series - 2 Probability 12 10 8 6 4 2 625 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 11 / 21

Truncated Fourier Series - 2 Probability 12 10 8 6 4 2 625 terms 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Probability 12 10 8 6 4 2 1250 term s 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 11 / 21

The Postulates 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. Jerry Gilfoyle The Rules of the Quantum Game 12 / 21

The Postulates 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. Jerry Gilfoyle The Rules of the Quantum Game 12 / 21

The Postulates 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. 3 The state of a system is represented by a wave function Ψ that is continuous, differentiable and contains all possible information about the system. The average value of any observable A is A = all space Ψ Â Ψd r. The intensity is proportional to Ψ 2. Jerry Gilfoyle The Rules of the Quantum Game 12 / 21

The Postulates 1 Each physical, measurable quantity, A, has a corresponding operator, Â, that satisfies the eigenvalue equation  φ = aφ and measuring that quantity yields the eigenvalues of Â. 2 Measurement of the observable A leaves the system in a state that is an eigenfunction of Â. 3 The state of a system is represented by a wave function Ψ that is continuous, differentiable and contains all possible information about the system. The average value of any observable A is A = all space Ψ Â Ψd r. The intensity is proportional to Ψ 2. 4 The time development of the wave function is determined by Ψ( r, t) i = 2 t 2µ 2 Ψ( r, t) + V ( r)ψ( r, t) µ reduced mass. Jerry Gilfoyle The Rules of the Quantum Game 12 / 21

Probabilities of Different Final States 0.4 Rectangular Wave in a Square Well 0.3 a = 1.0 A x 0 = 0.3 A x 1 = 0.5 A Probability 0.2 0.1 0.0 0 500 1000 1500 2000 2500 3000 Energy (ev) Jerry Gilfoyle The Rules of the Quantum Game 13 / 21

Probabilities of Different Final States - 2 1 Rectangular Wave in a Square Well a = 1.0 A 0.100 x 0 = 0.3 A x 1 = 0.5 A log(probability) 0.010 0.001 10-4 0 2000 4000 6000 8000 10 000 12 000 14 000 Energy (ev) Jerry Gilfoyle The Rules of the Quantum Game 14 / 21

Probabilities of Different Final States - 3 1 Rectangular Wave in a Square Well 0.100 log(probability) 0.010 0.001 10-4 10-5 0 20 40 60 80 100 Energy Level Jerry Gilfoyle The Rules of the Quantum Game 15 / 21

Probabilities of Different Final States - 4 1 Rectangular Wave in a Square Well 0.100 log(probability) 0.010 0.001 10-4 10-5 0 20 40 60 80 100 Energy Level Jerry Gilfoyle The Rules of the Quantum Game 16 / 21

Why a page limit? Nature 129, 312 (27 February 1932) Jerry Gilfoyle The Rules of the Quantum Game 17 / 21

Nuclear Fusion! Consider a case of one dimensional nuclear fusion. A neutron is in the potential well of a nucleus that we will approximate with an infinite square well with walls at x = 0 and x = L. The eigenfunctions and eigenvalues are E n = n2 2 π 2 2 ( nπx ) 2ma 2 φ n = a sin 0 x a a = 0 x < 0 and x > a. The neutron is in the n = 4 state when it fuses with another nucleus that is the same size, instantly putting the neutron in a new infinite square well with walls at x = 0 and x = 2a. 1 What are the new eigenfunctions and eigenvalues of the fused system? 2 What is the spectral distribution? Jerry Gilfoyle The Rules of the Quantum Game 18 / 21

Spectral Distribution for One-Dimensional Nuclear Fusion 0.6 Nuclear Fusion 0.5 0.4 bn 2 0.3 0.2 0.1 0.0 0 50 100 150 200 E n (units of E 1 ) Jerry Gilfoyle The Rules of the Quantum Game 19 / 21

Spectral Distribution for One-Dimensional Nuclear Fusion 0.6 Nuclear Fusion 0.5 0.4 bn 2 0.3 0.2 0.1 0.0 0 200 400 600 800 1000 E n (units of E 1 ) Jerry Gilfoyle The Rules of the Quantum Game 20 / 21

Spectral Distribution for One-Dimensional Nuclear Fusion 1 0.100 Nuclear Fusion Only non-zero values b n =0 for n even, except n=8 0.010 bn 2 0.001 10-4 10-5 0 200 400 600 800 1000 E n (units of E 1 ) Jerry Gilfoyle The Rules of the Quantum Game 21 / 21