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This document provides a continuation of the discussion on path integral approaches from a previous document. It begins by defining the matrix elements of the time evolution operator and explains how they give the probability amplitudes for a particle to transition between different states over time. It then shows how the time evolution operator and its matrix elements can be decomposed into multiple steps involving integrals over possible particle paths. The document derives expressions for the matrix elements and probability amplitudes involving position, momentum, and wavefunction terms. It provides equations for the Hamiltonian and time evolution operator in terms of these quantities. Finally, it takes the limit of many short time steps to express the probability amplitude as a continuous path integral over time.
![[stopped: pp. IR(23), skip notes: hawking radiation ref: townsend jacket f]
Ref’s
[1]Merzbacher, E., Quantum Mechanics, 2nd edition, 1970, Wiley & Sons, Inc.
[2] Baal, P., A COURSE IN FIELD THEORY,
http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html
[3]Cardy, J., Introduction to Quantum Field Theory
[4]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory](https://image.slidesharecdn.com/summerp62016update3slideshare-sqd-160606095618/85/Summerp62016update3-slideshare-sqd-7-320.jpg)
Summerp62016update3 slideshare sqd
- 1. Summer Physics 2016: Attachment2(part 1) for the Third Update Roa, F. J. P. Let us refresh ourselves with the basic approaches in Path Integral and this present document serves as a continuation of Summer Physics 2016: Attachment1 for the Second Update. To start, say we are given with the matrix elements ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ of the time evolution operator (1) 𝑈( 𝑇, 0) = 𝑈( 𝑇) = 𝑒𝑥𝑝(− 𝑖 ℏ ∫ 𝑑𝑡 𝐻̂(𝑡) 𝑇 0 ) These matrix elements are in terms of the continuous bases | 𝑥⟩ and these matrices bear the physical interpretation that these give the probability amplitudes that a particle initially at some initial state x at an initial time 𝑡 = 0 will be found at some other state 𝑥′ at a later time 𝑇 > 0. That is, from the initial state vector | 𝑥⟩ given at an initial time say, 𝑡 = 0, with the application of time evolution operator (1) we evolve this initial state vector into some other state vector | 𝜓⟩ = 𝑈( 𝑇)| 𝑥⟩ given at the later time T. Then we project this evolved state vector onto some other arbitrary state vector | 𝑥′⟩ to form the matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ that gives the cited probability amplitude. The time evolution operator is known to observe the causality principle and owing to this principle we can actually decompose (with a property of a group) a given time evolution operator as product of individual time evolution operators say for instance, (2) 𝑈( 𝑇, 𝑡1) = 𝑈( 𝑇, 𝑡2) 𝑈( 𝑡2, 𝑡1) also assuming that these operators are unitary. (3) 𝑈† 𝑈 = 𝑈𝑈† = 1
- 2. Given (2), we may write or decompose a particular matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ also as product of matrix elements (4) ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫ 𝑑𝑥1 ⟨ 𝑥′| 𝑈( 𝑇, 𝑡1)| 𝑥1⟩⟨ 𝑥1| 𝑈(𝑡1,0)| 𝑥⟩ also by having to insert the completeness relation of the bases (5) 1 = ∫| 𝑥1⟩ 𝑑𝑥1⟨ 𝑥1| Repeating this decomposition in an appropriate number of times inside the integral of (4) we would end up writing this matrix element as multiple integrations with the corresponding matrix elements (6.1) ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫∏ 𝑑𝑥 𝑖 𝑁−1 𝑖=1 ∏⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ 𝑁−1 𝑗 =0 In this we are making the following identifications (6.2) 𝑥 𝑁 = 𝑥′ , 𝑥0 = 𝑥 𝑡 𝑁 = 𝑇, 𝑡0 = 0 This resulting expression has a physical interpretation that the matrix element ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ that gives final probability amplitude is the integration over those paths each of which goes with its own probability amplitude. That is, this final probability amplitude is being decomposed into numerous probability amplitudes for various possible paths that a particle can go along and then summing or integrating these up to give the said final probability amplitude represented by the said matrix element. We make use of the fact that the projection of a given momentum state vector |𝑝𝑗⟩ onto an arbitrary position state vector |𝑥𝑗+1⟩ can be expressed as a wave packet (7.1)
- 3. ⟨𝑥𝑗+1|𝑝𝑗 ⟩ = 1 √2𝜋ℏ 𝑒 𝑖 𝑥 𝑗+1 𝑝 𝑗/ ℏ so each matrix element ⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ can be written using this wavepacket in the bases of momentum space (7.2) ⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗 )|𝑥𝑗⟩ = ∫ 𝑑𝑝𝑗 ⟨𝑥𝑗+1|𝑝𝑗 ⟩⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ also in this, it is assumed that the momentum bases satisfy the completeness relation (7.3) 1 = ∫|𝑝𝑗 ⟩𝑑𝑝𝑗⟨𝑝𝑗 | Say, considering that the Hamiltonian does not contain a time-dependent source so that we may write the time-evolution operator as (7.4) 𝑈(𝑡𝑗+1, 𝑡𝑗 ) = 𝑒𝑥𝑝(− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗)𝐻̂) The system Hamiltonian can be obtained given the system Lagrangian L through Legendre transformation (7.5.1) 𝐻 = π𝑥̇ − 𝐿 = 1 2 𝑚𝑥̇ 2 + 1 2 𝑚𝜔2 𝑥2 ̇ (7.5.2) 𝑝 = 𝑚𝑥̇ = π = 𝛿𝐿 𝛿𝑥̇ (7.5.3) 𝐻 = 1 2𝑚 𝑝2 + 1 2 𝑚𝜔2 𝑥2 We can use the matrix elements ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ in expressing ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ as
- 4. (7.5.4) ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = ∫⟨𝑝𝑗 |𝑈(𝑡𝑗+1 , 𝑡𝑗)|𝑝 𝑘⟩𝑑𝑝 𝑘 ⟨𝑝 𝑘|𝑥𝑗⟩ = 1 √2𝜋ℏ ∫ 𝑑𝑝 𝑘 ⟨ 𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ 𝑒− 𝑖 𝑥 𝑗 𝑝 𝑘/ ℏ Note that ⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ are the matrix elements of the time evolution operator (7.4) using the momentum bases and in these bases we can have the eigenvalue equation (7.5.5) 𝑒𝑥𝑝(− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗)𝐻̂) | 𝑝 𝑘⟩ = 𝑒𝑥𝑝 (− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘)| 𝑝 𝑘⟩ where the eigenvalues 𝐻 𝑘 of the Hamiltonian operator 𝐻̂ consist of the eigenvalues of the momentum and position operators (7.5.6) 𝐻 𝑘 = 1 2𝑚 𝑝 𝑘 2 + 1 2 𝑚𝜔2 𝑥 𝑘 2 Let us also add the orthonormality condition on these (continuous) momentum bases (7.5.7) ⟨𝑝𝑗 |𝑝 𝑘⟩ = 𝛿 ( 𝑝𝑗 − 𝑝 𝑘 ) Thus, writing the matrix elements ⟨𝑝𝑗 |𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ as (7.5.8) ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑝 𝑘⟩ = 𝑒𝑥𝑝(− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗)𝐻 𝑘) 𝛿 ( 𝑝𝑗 − 𝑝 𝑘 ) These are the needed matrix elements in (7.5.4) so integrating over 𝑝 𝑘 there yields, (7.5.9) ⟨𝑝𝑗|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = 1 √2𝜋ℏ 𝑒− 𝑖 𝑥 𝑗 𝑝 𝑗/ ℏ 𝑒𝑥𝑝(− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗)𝐻𝑗) In turn we use this in the matrix element (7.2) to write that as
- 5. (7.5.10) ⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = 1 2𝜋ℏ ∫ 𝑑𝑝𝑗 𝑒 𝑖( 𝑥 𝑗+1− 𝑥 𝑗 ) 𝑝 𝑗/ ℏ 𝑒𝑥𝑝(− 𝑖 ℏ (𝑡𝑗+1 − 𝑡𝑗 )( 1 2𝑚 𝑝𝑗 2 + 1 2 𝑚𝜔2 𝑥𝑗 2 )) which involves a Gaussian integration over the variable 𝑝𝑗 . Let us note that in some convenient (somewhat conventional) manner time is discretized in equal intervals such as (7.6) 𝑡𝑗+1 − 𝑡𝑗 ℏ = 𝑇 𝑁ℏ ∆𝑡 = 𝑇 𝑁 = 𝑡𝑗+1 − 𝑡𝑗 Of important use in this Gaussian integration is the integral result (7.7) ∫ 𝑑𝑦 𝑒 𝑎𝑦2 +𝑏𝑦 = √ 𝜋 −𝑎 𝑒−𝑏2 /4𝑎 ∞ −∞ so consequently, (7.5.10) results into (7.8) ⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ = √ 𝑚 𝑖2𝜋ℏ∆𝑡 𝑒𝑥𝑝( 𝑖∆𝑡 ℏ ( 𝑚 2 (𝑥𝑗+1 − 𝑥𝑗 ) 2 ∆𝑡2 − 𝑉(𝑥𝑗))) 𝑉(𝑥𝑗) = 1 2 𝑚𝜔2 𝑥𝑗 2 We could have stopped at (7.5.10) and utilize this result to express (6.1) in momentum space but we have opted to continue a little further at (7.8) intently so we may use this result in the partial product indicated in (6.1) and write this directly in coordinate space. We can write the following partial product
- 6. (7.9) ∏ √ 𝑚 𝑖2𝜋ℏ∆𝑡 𝑁−1 𝑗 =0 = ( 𝑚 𝑖2𝜋ℏ∆𝑡 ) 𝑁/2 so we may write the indicated partial product in (6.1) as (7.10) ∏⟨𝑥𝑗+1|𝑈(𝑡𝑗+1, 𝑡𝑗)|𝑥𝑗⟩ 𝑁−1 𝑗=0 = ( 𝑚 𝑖2𝜋ℏ∆𝑡 ) 𝑁/2 𝑒𝑥𝑝(∑ 𝑖∆𝑡 ℏ ( 𝑚 2 (𝑥𝑗+1 − 𝑥𝑗 )2 ∆𝑡2 − 𝑉(𝑥𝑗)) 𝑁−1 𝑗 =0 ) So in coordinate space we can write for the probability amplitude represented by (6.1) in path integral formalism and in coordinate space this can be expressed as (7.11) ⟨ 𝑥′| 𝑈(𝑇)| 𝑥⟩ = ∫ ( 𝑚 𝑖2𝜋ℏ∆𝑡 ) 𝑁/2 ∏ 𝑑𝑥 𝑘 𝑁−1 𝑘=1 𝑒𝑥𝑝( 𝑖 ℏ ∑ ∆𝑡 ( 𝑚 2 (𝑥𝑗+1 − 𝑥𝑗 )2 ∆𝑡2 − 𝑉(𝑥𝑗)) 𝑁−1 𝑗 =0 ) There could be an infinite number of paths that a particle can go from its initial state to the final state so we take this path integration in the limit as 𝑁 → ∞ and in this limit, we may take it that (7.12) lim 𝑁 → ∞ ∆𝑡 = 𝑇 𝑁 = 𝑑𝑡 so, the discrete case may pass into the continuous path integration over a continuous time integral although we retain the discretization ∆𝑡 = 𝑇 𝑁 in the measure (7.13) 𝔇𝑥 = ( 𝑚 𝑖2𝜋ℏ∆𝑡 ) 𝑁/2 ∏ 𝑑𝑥 𝑘 𝑁−1 𝑘=1 but we can always resort to its normalized form.
- 7. [stopped: pp. IR(23), skip notes: hawking radiation ref: townsend jacket f] Ref’s [1]Merzbacher, E., Quantum Mechanics, 2nd edition, 1970, Wiley & Sons, Inc. [2] Baal, P., A COURSE IN FIELD THEORY, http://www.lorentz.leidenuniv.nl/~vanbaal/FTcourse.html [3]Cardy, J., Introduction to Quantum Field Theory [4]Gaberdiel, M., Gehrmann-De Ridder, A., Quantum Field Theory