The Postulates What is a postulate? 1 suggest or assume the - PowerPoint PPT Presentation

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 / 20

The Postulates (the Rules of the Game) Each physical, measurable quantity, A , has a corresponding operator, ˆ A , that 1 satisfies the eigenvalue equation ˆ A φ = a φ and measuring that quantity yields the eigenvalues of ˆ A . Measurement of the observable A leaves the system in a state that is an 2 eigenfunction of ˆ A . The state of a system is represented by a wave function Ψ that is continuous, 3 differentiable and contains all possible information about the system. The ‘intensity’ is proportional to | Ψ | 2 and is interpreted as a probability. The average all space Ψ ∗ ˆ � value of any observable A is � A � = A Ψ d � r . The time and spatial dependence of Ψ( x , t ) is determined by the time dependent 4 Schroedinger equation. ∂ t Ψ( x , t ) = − � 2 ∂ 2 i � ∂ ∂ x 2 Ψ( x , t ) + V ( x )Ψ( x , t ) µ ≡ reduced mass. 2 µ Jerry Gilfoyle The Rules of the Quantum Game 2 / 20

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? V(x) What is the general solution to the previous question? What are the boundary conditions the solution must satisfy? What is the particular solution for x 0 a this potential? What is the energy of the particular solution? Jerry Gilfoyle The Rules of the Quantum Game 3 / 20

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 4 / 20

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 a φ ( 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 ∗ φ ∗ � φ n | ψ � = n ψ dx and � φ n | φ n � = n φ n dx = 1 −∞ −∞ 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. Jerry Gilfoyle The Rules of the Quantum Game 5 / 20

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. 1 Ψ( x , 0) = x 0 < x < x 1 and d = x 1 − x 0 √ d = 0 otherwise What possible values are obtained in an energy V(x) measurement? Ψ (x) What eigenfunctions contribute to this wave packet and what are their probabilities? x What will many measurements 0 x x a 0 1 of the energy give? Jerry Gilfoyle The Rules of the Quantum Game 6 / 20

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. Energy Levels n 1 Ψ( x , 0) = x 0 < x < x 1 and d = x 1 − x 0 √ 10 d = 0 otherwise 9 8 What possible values are Energy 7 obtained in an energy 6 V(x) measurement? 5 Ψ (x) What eigenfunctions contribute 4 3 to this wave packet and what 2 1 are their probabilities? x What will many measurements 0 x x a 0 1 of the energy give? Jerry Gilfoyle The Rules of the Quantum Game 6 / 20

L’Hˆ opital’s Rule If x → c f ( x ) = lim lim x → c g ( x ) = 0 or x → c f ( x ) = lim lim x → c g ( x ) = ±∞ and f ′ ( x ) lim g ′ ( x ) x → c exists, then f ( x ) f ′ ( x ) lim g ( x ) = lim . g ′ ( x ) x → c x → c Jerry Gilfoyle The Rules of the Quantum Game 7 / 20

Truncated Fourier Series - 1 12 10 1 term Probability 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1 12 12 10 5 terms 10 1 term Probability 8 Probability 8 6 6 4 4 2 2 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Position Position Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1 12 12 10 5 terms 10 1 term Probability 8 Probability 8 6 6 4 4 2 2 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Position Position 12 10 25 terms Probability 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 1 12 12 10 5 terms 10 1 term Probability 8 Probability 8 6 6 4 4 2 2 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Position Position 12 12 10 125 terms 10 25 terms Probability 8 Probability 8 6 6 4 4 2 2 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Position Position Jerry Gilfoyle The Rules of the Quantum Game 8 / 20

Truncated Fourier Series - 2 12 10 625 terms Probability 8 6 4 2 0 0.0 0.2 0.4 0.6 0.8 1.0 Position Jerry Gilfoyle The Rules of the Quantum Game 9 / 20

Truncated Fourier Series - 2 12 12 10 1250 term s 10 625 terms Probability 8 Probability 8 6 6 4 4 2 2 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Position Position Jerry Gilfoyle The Rules of the Quantum Game 9 / 20

Probabilities of Different Final States Rectangular Wave in a Square Well 0.4 a = 1.0 Å x 0 = 0.3 Å 0.3 x 1 = 0.5 Å Probability 0.2 0.1 0.0 0 500 1000 1500 2000 2500 3000 Energy ( eV ) Jerry Gilfoyle The Rules of the Quantum Game 10 / 20

Probabilities of Different Final States - 2 Rectangular Wave in a Square Well 1 a = 1.0 Å x 0 = 0.3 Å 0.100 x 1 = 0.5 Å log ( Probability ) 0.010 0.001 10 - 4 0 2000 4000 6000 8000 10000 12000 14000 Energy ( eV ) Jerry Gilfoyle The Rules of the Quantum Game 11 / 20

Probabilities of Different Final States - 3 Rectangular Wave in a Square Well 1 a = 1 . 0 ˚ A 0.100 x 0 = 0 . 3 ˚ A x 1 = 0 . 5 ˚ A 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 12 / 20

Probabilities of Different Final States - 4 Rectangular Wave in a Square Well 1 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 13 / 20

Why a page limit? Nature 129, 312 (27 February 1932) Jerry Gilfoyle The Rules of the Quantum Game 14 / 20

Width of a distribution? 0.20 0.15 2 0.10 | b n 0.05 0.00 0 50 100 150 200 k n ( a.u. ) Jerry Gilfoyle The Rules of the Quantum Game 15 / 20

The Quantum Program in One Dimension - So Far Solve the Schroedinger equation to get eigenfunctions and eigenvalues. 1 − � 2 ∂ 2 φ ( x ) + V φ ( x ) = E n φ ( x ) ∂ x 2 2 m For an initial wave packet ψ ( x ) use the completeness of the eigenfunctions. 2 ∞ � | ψ ( x ) � = b n | φ ( x ) � n =1 Apply the orthonormality � φ m | φ n � = δ m , n . 3 � ∞ � ∞ � � � ∞ � � � φ m | ψ � = � φ m | b n | φ � = b m = φ ∗ b n | φ � dx m −∞ n =1 n =1 Get the probability P n for measuring E n from | ψ � . 4 P n = | b n | 2 Do the free particle solution. 5 Put in the time evolution. 6 Jerry Gilfoyle The Rules of the Quantum Game 16 / 20

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 = a . The eigenfunctions and eigenvalues are E n = n 2 � 2 π 2 � 2 � n π x � φ n = a sin 0 ≤ x ≤ a 2 ma 2 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 = 2 a . 1 What are the new eigenfunctions and eigenvalues of the fused system? 2 What is the spectral distribution? 3 What is the average energy? Use the b n ’s. Jerry Gilfoyle The Rules of the Quantum Game 17 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion Nuclear Fusion 0.6 0.5 0.4 2 0.3 | b n 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 18 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion Nuclear Fusion 0.6 0.5 0.4 2 0.3 | b n 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 19 / 20

Spectral Distribution for One-Dimensional Nuclear Fusion Nuclear Fusion 1 Only non - zero values b n = 0 for n even, except n = 8 0.100 0.010 2 | b n 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 20 / 20