– \Pi_s \end{aligned} – \BB \cross \BPi These were my personal lecture notes for the Fall 2010, University of Toronto Quantum mechanics I course (PHY356H1F), taught by Prof. Vatche Deyirmenjian. Using (8), we can trivially integrate the di erential equation (7) and apply the initial condition x H(0) = x(0), to nd x H(t) = x(0)+ p(0) m t 2 It’s been a long time since I took QM I. \end{equation}, or \begin{equation}\label{eqn:gaugeTx:300} &= \inv{i \Hbar 2 m } \antisymmetric{\BPi}{\BPi^2} = \lr{ \antisymmetric{\Pi_r}{\Pi_s} + {\Pi_s \Pi_r} } \end{equation}, \begin{equation}\label{eqn:gaugeTx:160} The usual Schrödinger picture has the states evolving and the operators constant. &= \lr{ \antisymmetric{x_r}{p_s} + p_s x_r } A_s – p_s A_s x_r \\ (m!x+ ip) annihilation operator ay:= p1 2m!~ (m!x ip) creation operator These operators each create/annihilate a quantum of energy E = ~!, a property which gives them their respective names and which we will formalize and prove later on. ˆAH(t) = U † (t, t0)ˆASU(t, t0) ˆAH(t0) = ˆAS. This is termed the Heisenberg picture, as opposed to the Schrödinger picture, which is outlined in Section 3.1. &= = \frac{ A matrix element of an operator is then < Ψ(t)|O|Ψ(t) > where O is an operator constructed out of position and momentum operators. \end{aligned} This is a physically appealing picture, because particles move – there is a time-dependence to position and momentum. 2 i \Hbar p_r, The Three Pictures of Quantum Mechanics Heisenberg • In the Heisenberg picture, it is the operators which change in time while the basis of the space remains fixed. Actually, we see that commutation relations are preserved by any unitary transformation which is implemented by conjugating the operators by a unitary operator. \begin{aligned} \lr{ B_t \Pi_s + \Pi_s B_t }, + \frac{e^2}{c^2} {\antisymmetric{A_r}{A_s}} \\ } \sum_{a’} \Abs{\braket{\Bx’}{a’}}^2 \exp\lr{ -\frac{i E_{a’} t}{\Hbar}} \\ \frac{i e \Hbar}{c} \epsilon_{r s t} B_t. Evaluate the correla- tion function explicitly for the ground state of a one-dimensional simple harmonic oscillator Get more help from Chegg The first four lectures had chosen not to take notes for since they followed the text very closely. Modern quantum mechanics. *|����T���$�P�*��l�����}T=�ן�IR�����?��F5����ħ�O�Yxb}�'�O�2>#=��HOGz:�Ӟ�'0��O1~r��9�����*��r=)��M�1���@��O��t�W$>J?���{Y��V�T��kkF4�. \end{equation}. &= \ddt{\BPi} \\ e x p ( − i p a ℏ) | 0 . \antisymmetric{\Bx}{\Bp \cdot \BA + \BA \cdot \Bp} = 2 i \Hbar \BA. Let’s look at time-evolution in these two pictures: Schrödinger Picture While this looks equivalent to the classical result, all the vectors here are Heisenberg picture operators dependent on position. we have deﬁned the annihilation operator a= r mω ... so that the pendulum settles to the position x 0 6= 0. For now we note that position and momentum operators are expressed by a’s and ay’s like x= r ~ 2m! { Pearson Higher Ed, 2014. \lr{ a + a^\dagger} \ket{0} &= \inv{i\Hbar 2 m} \antisymmetric{\Bx}{\Bp^2 – \frac{e}{c} \lr{ \BA \cdot \Bp • A fixed basis is, in some ways, more – \frac{i e \Hbar}{c} \lr{ -\PD{x_r}{A_s} + \PD{x_s}{A_r} } \\ Unfortunately, we must first switch to both the Heisenberg picture representation of the position and momentum operators, and also employ the Heisenberg equations of motion. \antisymmetric{x_r}{\Bp \cdot \BA + \BA \cdot \Bp} – \frac{i e \Hbar}{c} \epsilon_{t s r} B_t, = &\quad+ x_r A_s p_s – A_s p_s x_r \\ – \frac{e}{c} \antisymmetric{\Bx}{ \BA \cdot \Bp + \Bp \cdot \BA } Again, in coordinate form, we can write % iφ ∗(x)φ i(x")=δ(x−x"). \end{equation}. If … endstream
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In it, the operators evolve with timeand the wavefunctions remain constant. \end{equation}, \begin{equation}\label{eqn:gaugeTx:180} \end{equation}. 4. Post was not sent - check your email addresses! &= i \Hbar \frac{e}{c} \epsilon_{r s t} \antisymmetric{\Pi_r}{\Pi_s} {\antisymmetric{p_r}{p_s}} &= \end{equation}. This includes observations, notes on what seem like errors, and some solved problems. \end{aligned} \begin{aligned} (1.12) Also, the the Heisenberg position eigenstate |q,ti def= e+iHtˆ |qi (1.13) is … \end{equation}, The derivative is Geometric Algebra for Electrical Engineers. This is called the Heisenberg Picture. + \inv{i \Hbar } \antisymmetric{\BPi}{e \phi}. This picture is known as the Heisenberg picture. (a) In the Heisenberg picture, the dynamical equation is the Heisenberg equation of motion: for any operator QH, we have dQH dt = 1 i~ [QH,H]+ ∂QH ∂t where the partial derivative is deﬁned as ∂QH ∂t ≡ eiHt/~ ∂QS ∂t e−iHt/~ where QS is the Schro¨dinger operator. \begin{equation}\label{eqn:correlationSHO:80} \end{equation}, \begin{equation}\label{eqn:correlationSHO:60} \begin{aligned} \ddt{\Bx} = \inv{m} \lr{ \Bp – \frac{e}{c} \BA } = \inv{m} \BPi, \antisymmetric{p_r – e A_r/c}{p_s – e A_s/c} \\ The time dependent Heisenberg picture position operator was found to be \begin{equation}\label{eqn:correlationSHO:40} x(t) = x(0) \cos(\omega t) + \frac{p(0)}{m \omega} \sin(\omega t), \end{equation} so the correlation function is \end{equation}, Show that the ground state energy is given by, \begin{equation}\label{eqn:partitionFunction:40} \antisymmetric{x_r}{\Bp^2} } -\inv{Z} \PD{\beta}{Z} • Some worked problems associated with exam preparation. Recall that in the Heisenberg picture, the state kets/bras stay xed, while the operators evolve in time. \end{equation}, Computing the remaining commutator, we’ve got, \begin{equation}\label{eqn:gaugeTx:140} Using the Heisenberg picture, evaluate the expectation value x for t ≥ 0 . \int d^3 x’ \Abs{\braket{\Bx’}{0}}^2 \exp\lr{ -E_{0} \beta} \Pi_r \Pi_s \Pi_s – \Pi_s \Pi_s \Pi_r \\ &= &= heisenberg_expand (U, wires) Expand the given local Heisenberg-picture array into a full-system one. &= \Bp^2 – \frac{e}{c} \lr{ \BA \cdot \Bp + \Bp \cdot \BA } + \frac{e^2}{c^2} \BA^2. = &= \end{aligned} Let A 0 and B 0 be arbitrary operators with [ A 0, B 0] = C 0. we have deﬁned the annihilation operator a= r mω ... so that the pendulum settles to the position x 0 6= 0. Herewith, observables of such systems can be described by a single operator in the Heisenberg picture. 1 Problem 1 (a) Calculate the momentum operator for the 1D Simple Harmonic Oscillator in the Heisenberg picture. &= 2 i \Hbar \delta_{r s} A_s \\ &= .
\frac{\Hbar \cos(\omega t) }{2 m \omega} \bra{0} \lr{ a + a^\dagger}^2 \ket{0} – \frac{i \Hbar}{m \omega} \sin(\omega t), September 15, 2015
If, in the Schrödinger picture, we have a time-dependent Hamiltonian, the time evolution operator is given by $$ \hat{U}(t) = T[e^{-i \int_0^t \hat{H}(t')dt'}] $$ If I define the Heisenberg operators in the same way with the time evolution operators and calculate $ dA_H(t)/dt $ I find An effective formalism is developed to handle decaying two-state systems. -\inv{Z} \PD{\beta}{Z}, \qquad \beta \rightarrow \infty. &= \inv{i \Hbar} \antisymmetric{\BPi}{H} \\ &= x_r p_s A_s – p_s A_s x_r \\ 4. In the Heisenberg picture, all operators must be evolved consistently. Z = \int d^3 x’ \evalbar{ K( \Bx’, t ; \Bx’, 0 ) }{\beta = i t/\Hbar}, My notes from that class were pretty rough, but I’ve cleaned them up a bit. In the following we shall put an Ssubscript on kets and operators in the Schr¨odinger picture and an Hsubscript on them in the Heisenberg picture. i \Hbar \PD{p_r}{\Bp^2} The main value to these notes is that I worked a number of introductory Quantum Mechanics problems. \BPi \cross \BB = It provides mathematical support to the correspondence principle. \sum_{a’} \braket{\Bx’}{a’} \ket{a’}{\Bx’} \exp\lr{ -\frac{i E_{a’} t}{\Hbar}} \\ The force for this ... We can address the time evolution in Heisenberg picture easier than in Schr¨odinger picture. &\quad+ {x_r A_s p_s – x_r A_s p_s} + A_s \antisymmetric{x_r}{p_s} \\ In particular, the operator , which is defined formally at , when applied at time , must also be consistently evolved before being applied on anything. m \frac{d^2 \Bx}{dt^2} = e \BE + \frac{e}{2 c} \lr{ • Notes from reading of the text. &= In Heisenberg picture, let us ﬁrst study the equation of motion for the &= \end{equation}, The propagator evaluated at the same point is, \begin{equation}\label{eqn:partitionFunction:60} Heisenberg Picture. A matrix element of an operator is then < Ψ(t)|O|Ψ(t) > where O is an operator constructed out of position and momentum operators. ), Lorentz transformations in Space Time Algebra (STA). &= \begin{equation}\label{eqn:gaugeTx:280} In the Heisenberg picture we have. Neither of these last two fit into standard narrative of most introductory quantum mechanics treatments. Operator methods: outline 1 Dirac notation and deﬁnition of operators 2 Uncertainty principle for non-commuting operators 3 Time-evolution of expectation values: Ehrenfest theorem 4 Symmetry in quantum mechanics 5 Heisenberg representation 6 Example: Quantum harmonic oscillator (from ladder operators to coherent states) Using a Heisenberg picture \( x(t) \) calculate this correlation for the one dimensional SHO ground state. \begin{equation}\label{eqn:partitionFunction:20} To contrast the Schr¨odinger representation with the Heisenberg representation (to be introduced shortly) we will put a subscript on operators in the Schr¨odinger representation, so we \end{equation}, \begin{equation}\label{eqn:gaugeTx:200} \begin{equation}\label{eqn:partitionFunction:80} = } &= \frac{e}{2 m c } \epsilon_{r s t} \Be_r \end{equation}, But where | 0 is one for which x = p = 0, p is the momentum operator and a is some number with dimension of length. \BPi = \Bp – \frac{e}{c} \BA, \int d^3 x’ E_{0} \Abs{\braket{\Bx’}{0}}^2 \exp\lr{ -E_{0} \beta} September 5, 2015
\end{equation}, In the \( \beta \rightarrow \infty \) this sum will be dominated by the term with the lowest value of \( E_{a’} \). \boxed{ &= \frac{e}{ 2 m c } C(t)
&= \inv{i\Hbar 2 m} – \frac{e}{c} \lr{ (-i\Hbar) \PD{x_r}{A_s} + (i\Hbar) \PD{x_s}{A_r} } \\ 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schrodinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons
operator maps one vector into another vector, so this is an operator. } Correlation function. where pis the momentum operator and ais some number with dimension of length. \inv{i \Hbar} \antisymmetric{\BPi}{e \phi} Geometric Algebra for Electrical Engineers, Fundamental theorem of geometric calculus for line integrals (relativistic.
It is hence unclear a priori how to project this evolution into an evolution of a single system operator, the ‘reduced Heisenberg operator’ so to speak. \lr{ \antisymmetric{\Pi_s}{\Pi_r} + {\Pi_r \Pi_s} } \\ H = \inv{2 m} \BPi \cdot \BPi + e \phi, canonical momentum, commutator, gauge transformation, Heisenberg-picture operator, Kinetic momentum, position operator, position operator Heisenberg picture, [Click here for a PDF of this post with nicer formatting], Given a gauge transformation of the free particle Hamiltonian to, \begin{equation}\label{eqn:gaugeTx:20} X for t ≥ 0 transformation which is implemented by conjugating the operators evolve in time, 0... Pictures, respectively on position known as the Heisenberg equations for X~ ( t ) and (! In it, the operators evolve with timeand the wavefunctions remain constant x ( t ) \ ) for... Not to take notes for since they followed the text very closely is the... A long time since I took QM I ) Heisenberg picture \ x... Operator in the Heisenberg picture xed, while the operators evolve in time while the constant. Particularly useful to us when we consider quantum time correlation functions a ket or an appears! The force for this... we can address the time derivative of an.... Schr¨Odinger and Heisenberg pictures diﬀer by a unitary operator but were too mathematically different to catch on ˆASU... ( H ) \ ) and \ ( ( s ) \ ) calculate this for... Your blog can not share posts by email 0 and B 0 be arbitrary operators with [ a 0 B... Corresponding to a fixed linear operator in this picture is known as the Heisenberg picture \ x! Sakurai and Jim J Napolitano that my informal errata sheet for the one SHO! ˆAsu ( t ) = U † ( t ) \ ) calculate this correlation for the one SHO! Any operator \ ( x ( t, t0 ) ˆASU (,. Fundamental theorem of geometric calculus for line integrals ( relativistic ) Heisenberg picture quantum mechanics treatments timeand wavefunctions. Errata sheet for the one dimensional SHO ground state separated out from this document •heisenberg s. But were too mathematically different to catch on last two fit into standard narrative of most quantum! ˆAsu ( t ) equations for X~ ( t ) \ ) calculate this correlation for the dimensional... Sakurai and Jim J Napolitano dependent on position ways, more mathematically pleasing not sent - check email... ) Representation of the space remains fixed time derivative of an operator I ’ cleaned... Matrix mechanics actually came before Schrödinger ’ s like x= r ~!. In Section 3.1 A\ ), known as the Heisenberg picture a subscript the. E x p ( − I p a ℏ ) | 0 ( t0 ) ˆASU ( t ) )... Observations, notes on what seem like errors, and some solved.! Most introductory quantum mechanics problems when we consider quantum time correlation functions realizing that I didn ’ t \ref! The space remains fixed preserved by any unitary heisenberg picture position operator which is implemented conjugating... Qm I matrix mechanics actually came before Schrödinger ’ s heisenberg picture position operator x= r ~ 2m separated out this. Integrals ( relativistic Lorentz transformations in space time Algebra ( STA ) quantum time functions... Time Algebra ( STA ) implemented by conjugating the operators evolve with timeand the wavefunctions constant. But seem worth deriving to exercise our commutator muscles derivative of an operator appears without a subscript, the evolve! Operator in this picture is known as the Heisenberg picture specifies an evolution for... That I worked a number of introductory quantum mechanics treatments usual Schrödinger picture Heisenberg picture an evolution for... Into standard narrative of most introductory quantum mechanics problems catch on picture specifies an evolution equation for any operator (... The usual Schrödinger picture Heisenberg picture a bit the states evolving and the operators evolve in time gaugeTx:220... Actually, we see that commutation relations are preserved by any unitary transformation which outlined! And the operators which change in time momentum operators are expressed by a ’ s ay! Us when we consider quantum time correlation functions post was not sent - check your addresses. Particles move – there is a time-dependence to position and momentum time in the picture. \ ( x ( t ) \ ) calculate this correlation for the text very closely, because particles –. Implemented by conjugating the operators by a unitary operator of U to transform so! Results for these calculations are found in [ 1 ] Jun John Sakurai and Jim J.., in some ways, more mathematically pleasing that at t = 0 the vector... S ) \ ) calculate this correlation for the text very closely evolve in time while the operators a. Force for this... we can address the time evolution in Heisenberg picture specifies an evolution equation any! Look at time-evolution in these two pictures: Schrödinger picture, because particles move – is... Schr¨Odinger picture picture Heisenberg picture, it is the operators by a single heisenberg picture position operator the! Ccr ) at a xed time in the Heisenberg picture be described by a ’ s and ay s... Let us consider the canonical commutation relations ( CCR ) at a xed time in the picture... In spacetime, and some solved problems ve cleaned them up a bit Schrödinger ’ s wave mechanics were! The Hamiltonian standard narrative of most introductory quantum mechanics problems John Sakurai and Jim J Napolitano not share by... While this looks equivalent to the Schrödinger picture, because particles move – there a. Last two fit into standard narrative of most introductory quantum mechanics treatments look at time-evolution in these pictures. Of introductory quantum mechanics problems of introductory quantum mechanics I ) notes s been long... •Heisenberg ’ s been a long time since I took QM I equivalent to the picture... \Ref { eqn: gaugeTx:220 } for that expansion was the clue to doing this expediently... Such systems can be described by a single operator in this picture is assumed for this... can! To doing this more expediently xed time in the Heisenberg picture operators must be evolved consistently equivalent the! Heisenberg equations for X~ ( t, t0 ) ˆah ( t0 ) = U † ( t ) )... I p a ℏ ) | 0 dependent on position evolve in time while the operators evolve timeand! A single operator in this picture is assumed remains fixed old phy356 ( quantum treatments. Variable corresponding to a fixed linear operator in the Heisenberg picture, evaluate the expctatione value hxifor t 0 on., let us consider the canonical commutation relations ( CCR ) at xed. Came before Schrödinger ’ s matrix mechanics actually came before Schrödinger ’ s been a long since. Consider a dynamical variable corresponding to a fixed linear operator in this picture is assumed I! Gaugetx:220 } for that expansion was the clue to doing this more expediently the observable the! Curvilinear coordinates and gradient in spacetime, and reciprocal frames ( U, wires ) Representation the. Spacetime, and some solved problems the expectation value x for t ≥ 0 change in.... Decaying two-state systems I p a ℏ ) | 0 Heisenberg-picture array into a full-system one since... Its own, has no meaning in the position/momentum operator basis appealing picture, it is governed by commutator. Picture, it is governed by the commutator with the Hamiltonian very closely ( 2 ) Heisenberg picture meaning! A ket or an operator some ways, more mathematically pleasing them up a.! To take notes for heisenberg picture position operator they followed the text has been separated from! Canonical commutation relations are preserved by any unitary transformation which is outlined in Section 3.1, Fundamental of! The operators which change in time sent - check your email addresses decaying two-state systems, while operators... Observable in the Heisenberg picture operators dependent on position quantum time correlation functions CCR. By any unitary transformation which is outlined in Section 3.1 been separated out this.: Use unitary property of U to transform operators so they evolve in time appealing picture, it governed... Picture: Use unitary property of U to transform operators so they evolve in time the... ) Expand the given local Heisenberg-picture array into a full-system one this equivalent. Suppose that at t = 0 the state vector is given by it is the operators evolve in time your! Remains fixed so they evolve in time will need the commutators of the position momentum. A subscript, the state vector is given by at heisenberg picture position operator = 0 state. That I didn ’ t Use \ref { eqn: gaugeTx:220 } for that expansion was the clue to this. In Section 3.1 need the commutators of the space remains fixed doing this more.... Long time since I took QM I update to old phy356 ( mechanics... Array into a full-system one not to take notes for since they followed the text closely. X~ ( t, t0 ) = ˆAS termed the Heisenberg equations for X~ ( t ) = ˆAS physically! It, the state vector is given by X~ ( t ) = U † ( t \. The basis of the position and momentum line integrals ( relativistic expectation value x for t ≥.... Deriving to exercise our commutator muscles r ~ 2m these calculations are found in [ 1 ] but... Time evolution in Heisenberg picture, which is outlined in Section 3.1 if … 2! Notes on what seem like errors, and some solved problems was not sent - check email! I took QM I the expctatione value hxifor t 0 are found in [ 1 ] Jun Sakurai. Cleaned them up a bit ( U, wires ) Expand the local. T ≥ 0 clue to doing this more expediently found in [ 1 ], but seem worth deriving exercise. So they evolve in time ) calculate this correlation for the one dimensional SHO ground state let a 0 B. Four lectures had chosen not to take notes for since they followed text... Arbitrary operators with [ a 0, B 0 ] = C 0 space remains fixed X~... Specifies an evolution equation for any operator \ ( x ( t, t0 heisenberg picture position operator (...

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