pictures in quantum mechanics
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QMPT 540
Pictures in Quantum Mechanics• Quick review (see Appendix A)
Schrödinger picture (usual)
• Schrödinger equation (SE) for many-particle state
• given at
• time-independent Hamiltonian
• with
• time-evolution operator in Schrödinger picture
|�S(t)� = |�(t)�
i� �
�t|�S(t)� = H |�S(t)�
|�S(t0)� t0
|�S(t)⇥ = US(t� t0) |�S(t0)⇥
US(t� t0) = exp�� i
�H(t� t0)⇥
QMPT 540
Heisenberg picture• Transform time dependence to operators while making state kets
“timeless” • Define
• It follows that
• and therefore
• For operators employ
• to obtain
• with
|�H(t)� = exp�
i
�Ht
⇥|�S(t)�
i� �
�t|�H(t)⇥ = �H |�H(t)⇥+ H |�H(t)⇥ = 0
|�H(t)⇥ � |�H⇥
OS |�S(t)� = |��S(t)�
|��H⇥ = exp
�i
�Ht
⇥|��
S(t)⇥
= exp�
i
�Ht
⇥OS exp
�� i
�Ht
⇥exp
�i
�Ht
⇥|�S(t)⇥ = OH(t) |�H⇥
OH(t) = exp�
i
�Ht
⇥OS exp
�� i
�Ht
⇥
QMPT 540
Equation of motion for Heisenberg operators• Use definition
• showing that if the Schrödinger operator commutes with Hamiltonian, the corresponding Heisenberg operator is constant of motion
i� ⇥
⇥tOH(t) =
�i� ⇥
⇥texp
�i
�Ht
⇥⇥OS exp
�� i
�Ht
⇥
+ exp�
i
�Ht
⇥OS
�i� ⇥
⇥texp
�� i
�Ht
⇥⇥
= �HOH(t) + OH(t)H =⇤OH(t), H
⌅
= exp�
i
�Ht
⇥ ⇤OS , H
⌅exp
�� i
�Ht
⇥
QMPT 540
Properties• Note that
• and
• For energy eigenkets
• and
• So
|�nS (t)� = e�iEnt/� |�n�
= e�iHt/� |�n�
|�n� = |�nH �
H |�n� = En |�n�
OS = OH(t = 0)
|�H� = |�S(t = 0)�
QMPT 540
Use definitions• Write in detail
• introducing appropriate completeness relations with exact eigenstates
G(�,⇥; t � t⇥) = � i
�
�⇤(t � t⇥)e
i� EN
0 (t�t⇥) ⇥�N0 | a�e�
i� H(t�t⇥)a†⇥ |�N
0 ⇤
�⇤(t⇥ � t)ei� EN
0 (t⇥�t) ⇥�N0 | a†⇥e�
i� H(t⇥�t)a� |�N
0 ⇤⇥
= � i
�
�⇤(t � t⇥)
⇤
m
ei� (EN
0 �EN+1m )(t�t⇥) ⇥�N
0 | a� |�N+1m ⇤ ⇥�N+1
m | a†⇥ |�N0 ⇤
�⇤(t⇥ � t)⇤
n
ei� (EN
0 �EN�1n )(t⇥�t) ⇥�N
0 | a†⇥ |�N�1n ⇤ ⇥�N�1
n | a� |�N0 ⇤
⇥
H |�N+1m � = EN+1
m |�N+1m �
H |�N�1n � = EN�1
n |�N�1n �
�+F = EN+1
0 � EN0 QMPT 540
Spectral functions• Physics of knock-out experiments to be discussed shortly can be
interpreted nicely using spectral functions • For the removal of particles, we have the hole spectral function
• with
• A similar addition probability density is available for adding particles (particle spectral function)
Sh(�;E) =1⇤
Im G(�,�;E) E ⇥ ⌅�F
=⇥
n
���⇤�N�1n | a� |�N
0 ⌅���2⇥(E � (EN
0 � EN�1n ))
Sp(�;E) = � 1⇤
Im G(�,�;E) E ⇥ ⌅+F
=⇥
m
���⇤�N+1m | a†� |�N
0 ⌅���2⇥(E � (EN+1
m � EN0 ))
��F = EN0 � EN�1
0
1E ± i⇥
= P 1E⇥ i⇤�(E)
QMPT 540
Occupation and depletion• Occupation number
• Depletion
• Obvious sum rule
n(�) = ⇥�N0 | a†�a� |�N
0 ⇤ =⇥
n
���⇥�N�1n | a� |�N
0 ⇤���2
=⇤ ⇥�F
�⇥dE
⇥
n
���⇥�N�1n | a� |�N
0 ⇤���2⇥(E � (EN
0 � EN�1n ))
=⇤ ⇥�F
�⇥dE Sh(�;E)
d(�) = ⇥�N0 | a�a†� |�N
0 ⇤ =⇥
m
���⇥�N+1m | a†� |�N
0 ⇤���2
=⇤ �
⇥+F
dE⇥
m
���⇥�N+1m | a†� |�N
0 ⇤���2⇥(E � (EN+1
m � EN0 ))
=⇤ �
⇥+F
dE Sp(�;E)
n(�) + d(�) = ��N0 | a†�a� |�N
0 ⇥ + ��N0 | a�a†� |�N
0 ⇥ = ��N0 |�N
0 ⇥ = 1
QMPT 540
Expectation values of operators in ground state• Consider one-body operator
• One-body density matrix element • can be obtained from sp propagator
• or
��N0 | O |�N
0 ⇥ =�
�,⇥
��| O |⇥⇥ ��N0 | a†�a⇥ |�N
0 ⇥ =�
�,⇥
��| O |⇥⇥n�⇥
n�⇥ � ⇥�N0 | a†�a⇥ |�N
0 ⇤
n⇥� =⇥
dE
2⌅ieiE⇤ G(�,⇥;E)
=⇥
dE
2⌅ieiE⇤
�
m
⇥�A0 | a� |�A+1
m ⇤ ⇥�A+1m | a†⇥ |�A
0 ⇤E � (EA+1
m � EA0 ) + i⇤
+⇥
dE
2⌅ieiE⇤
�
n
⇥�N0 | a†⇥ |�N�1
n ⇤ ⇥�N�1n | a� |�N
0 ⇤E � (EN
0 � EN�1n ) � i⇤
=�
n
⇥�N0 | a†⇥ |�N�1
n ⇤ ⇥�N�1n | a� |�N
0 ⇤ = ⇥�N0 | a†⇥a� |�N
0 ⇤
n⇥� =1⇤
� ⇤�F
�⇥dE Im G(�,⇥;E) = ��N
0 | a†⇥a� |�N0 ⇥
QMPT 540
Magic?!: energy sum rule• Consider
• Earlier results yield
• Insert
• Sum over
I� =1⇥
⇥ ⇥�F
�⇥dE E Im G(�,�;E) =
⇥ ⇥�F
�⇥dE E Sh(�;E)
=�
m
(EN0 � EN�1
m ) ⇥�N0 | a†� |�N�1
m ⇤ ⇥�N�1m | a� |�N
0 ⇤
= ⇥�N0 | a†�a�H |�N
0 ⇤ ��
m
⇥�N0 | a†�EN�1
m |�N�1m ⇤ ⇥�N�1
m | a� |�N0 ⇤
= ⇥�N0 | a†�a�H |�N
0 ⇤ � ⇥�N0 | a†�Ha� |�N
0 ⇤ = ⇥�N0 | a†�[a�, H] |�N
0 ⇤
[a�, H] =�
⇥
��| T |⇥⇥ a⇥ +�
⇥⇤⌅
(�⇥|V |⇤⌅)a†⇥a⌅a⇤
I� =�
⇥
��| T |⇥⇥ ��N0 | a†�a⇥ |�N
0 ⇥ +�
⇥⇤⌅
(�⇥|V |⇤⌅) ��N0 | a†�a†⇥a⌅a⇤ |�N
0 ⇥
��
�
I� = ��N0 | T |�N
0 ⇥ + 2 ��N0 | V |�N
0 ⇥
QMPT 540
Galitski-Migdal energy sum rule (Koltun)• Combine with half the expectation value of the kinetic energy
• complete result only when there are no three- or higher-body interactions
• sp propagator (hole part) yields energy of the ground state
• later: particle part yields elastic scattering cross section
EN0 = ⇤�N
0 | H |�N0 ⌅
=12⌅
⌃ ⇤�F
�⇥dE
⇧
�,⇥
{⇤�|T |⇥⌅ + E ⇤�,⇥} Im G(⇥,�;E)
=12
�
⇤⇧
�,⇥
⇤�|T |⇥⌅n�⇥ +⇧
�
⌃ ⇤�F
�⇥dE E Sh(�;E)
⇥
⌅
QMPT 540
Noninteracting propagator• Propagator for involves interaction picture
• with corresponding ground state
• as for IPM so closed-shell atom or nucleus for example • Operators
• assuming is diagonal in this basis
H0
G(0)(�,⇥; t � t�) = � i
� ⇤�N0 | T [a�I (t)a
†⇥I
(t�)] |�N0 ⌅
H0 |�N0 � = E�N
0|�N
0 �
E�N0
=�
�<F
��
a�I (t) = ei� H0ta�e�
i� H0t = e�i⇥�t/�a�
a†�I(t) = e
i� H0ta†�e�
i� H0t = ei⇥�t/�a†�
H0
QMPT 540
Evaluate noninteracting sp propagator• Insert
• propagation of a particle or a hole on top of noninteracting ground state
• directly:
• FT
G(0)(�,⇥; t� t⇥) = G(0)+ (�,⇥; t� t⇥) + G(0)
� (�,⇥; t� t⇥)
= � i
�⇤�⇥
�⌅(t� t⇥)⌅(�� F )e�
i� ⇤�(t�t�) � ⌅(t⇥ � t)⌅(F � �)e
i� ⇤�(t��t)
⇥
H0 a†� |�N0 � = (E�N
0+ ⇥�) a†� |�N
0 � � > F
H0 a� |�N0 ⇥ = (E�N
0� ⇥�) a� |�N
0 ⇥ � < F
G(0)(�,⇥;E) = ⇤�,⇥
�⇧(�� F )
E � ⌃� + i⌅+
⇧(F � �)E � ⌃� � i⌅
⇥
QMPT 540
Noninteracting spectral functions• Imaginary parts yield all the strength at one location
• in this basis: either completely full or empty
• other basis
S(0)h (�;E) =
1⌅
Im G(0)(�,�;E) E < ⇧(0)�
F
= ⇥(E � ⇧�) ⇤(F � �)
S(0)p (�;E) = � 1
⌅Im G(0)(�,�;E) E > ⇧(0)+
F
= ⇥(E � ⇧�) ⇤(�� F )
n(0)(�) =� ⇥(0)�
F
�⇥dE ⇥(E � ⌅�) ⇤(F � �) = ⇤(F � �)
G(0)(rms, r�m�
s;E) = ⇥�N0 | arms
1E � (H0 � E�N
0) + i⇥
a†r�m�s
|�N0 ⇤
+ ⇥�N0 | a†r�m�
s
1E � (E�N
0� H0) � i⇥
arms |�N0 ⇤
=⇤
�
�⇥rms|�⇤⇥�|r�m�
s⇤⇤(� � F )E � ⌅� + i⇥
+⇥rms|�⇤⇥�|r�m�
s⇤⇤(F � �)E � ⌅� � i⇥
⇥
QMPT 540
Direct knockout reactions• Atoms: (e,2e) reaction • Nuclei: (e,e’p) reaction [and others like (p,2p), (d,3He), (p,d), etc.]
• Physics: transfer large amount of momentum and energy to a bound particle; detect ejected particle together with scattered projectile → construct spectral function
• Simple analysis • Initial state: ground state
• Final state:
• Probe: acts as one-body excitation operator transferring momentum to a particle
• 2nd quantization (no spin)
|�i� = |�N0 �
|�f � = a†p |�N�1n �
�(q) =N�
j=1
exp (iq · rj)�q
�(q) =�
p,p�
⇥p| exp (iq · r) |p⇥⇤ a†pap� =�
p
a†pap��q
QMPT 540
Transition matrix element• Impulse approximation: struck particle is ejected
• Other assumption: final state ~ plane wave on top of N-1 particle eigenstate (more serious in practical experiments) but good approximation if ejectile momentum large enough
• Write
• last term FSI: interaction between ejected particle and others
• If relative momentum large enough, interaction can be neglected:
• PWIA = plane wave impulse approximation
⇥�f | ⇥(q) |�i⇤ =�
p�
⇥�N�1n | apa†p�ap���q |�N
0 ⇤
=�
p�
⇥�N�1n | �p�,pap���q + a†p�ap���qap |�N
0 ⇤
� ⇥�N�1n | ap��q |�N
0 ⇤
HN =N�
i=1
p2i
2m+
N�
i<j=1
V (i, j) = HN�1 +p2
N
2m+
N�1�
i=1
V (i, N)
QMPT 540
Cross section• Fermi’s Golden Rule
• with energy transfer linking initial
and final state energy • Define
• Rewrite knockout cross section
• More comprehensive treatment requires inclusion of FSI
d⇤ ⇥�
n
�(�⌅ + Ei � Ef )| ⇤�f | ⇥(q) |�i⌅ |2
�� Ei = EN0
Ef = EN�1n + p2/2m
pmiss = p� �q
Emiss = p2/2m� �� = EN0 � EN�1
n
d⇥ ⇥�
n
�(Emiss � EN0 + EN�1
n )| ⇤�N�1n | apmiss |�N
0 ⌅ |2
= Sh(pmiss;Emiss)
QMPT 540
(e,2e) data for atoms• Start with Hydrogen • Ground state wave function
• (e,2e) removal amplitude
⇥1s(p) =23/2
�
1(1 + p2)2
�0| ap |n = 1, ⇤ = 0⇥ = �p |n = 1, ⇤ = 0⇥ = �1s(p)
Hydrogen 1s wave function “seen” experimentally Phys. Lett. 86A, 139 (1981)
QMPT 540
Helium• IPM description is very successful • Closed-shell configuration
• Reaction more complicated than for Hydrogen
• DWIA (distorted wave impulse approximation)
agreement with IPM! → 1
1s2
Phys. Rev. A8, 2494 (1973)
S =⇥
dp����N�1
n | ap |�N0 ⇥
��2
QMPT 540
Other closed-shell atoms• Spectroscopic factor become less than 1 • Neon removal: S = 0.92 with two fragments each 0.04
• IPM not the whole story: fragmentation of sp strength
• Summed strength: like IPM • IPM wave functions still excellent
• Example: Argon S = 0.95
• Rest in 3 small fragments
2p
0 1 2 3 p (a.u.)
0.0
0.4
0.8
1.2
1.6
2.0
Diff
eren
tial c
ross
sect
ion
(10−
3 a.u
Argon
3p
3p
QMPT 540
Fragmentation in atoms• ~All the strength remains below (above) the Fermi energy in
closed-shell atoms • Fragmentation can be interpreted in terms of mixing between
• and
• with the same “global” quantum numbers
• Example: Argon ground state
• Ar+ ground state • excited state
• also
• and
a� |�N0 �
a�a⇥a†⇤ |�N0 �
|�N0 � = |(3s)2(3p)6(2s)2(2p)6(1s)2�
|(3p)�1� = a3p |�N0 � = |(3s)2(3p)5(2s)2(2p)6(1s)2�
|(3s)�1� = a3s |�N0 � = |(3s)1(3p)6(2s)2(2p)6(1s)2�
|(3p)�24s� = a3pa3pa†4s |�N
0 � = |(4s)1(3s)2(3p)4(2s)2(2p)6(1s)2�|(3p)�2nd� = a3pa3pa
†nd |�N
0 � = |(nd)1(3s)2(3p)4(2s)2(2p)6(1s)2�
QMPT 540
Argon spectroscopic factors• s strength also in the continuum: Ar++ + e • note vertical scale
• red bars: 3s fragments exhibit substantial fragmentation
8%
QMPT 540
(e,e’p) data for nuclei• Requires DWIA • Distorted waves required to describe elastic proton scattering at
the energy of the ejected proton
• Consistent description requires that cross section at different energy for the outgoing proton is changed accordingly
• Requires substantial beam energy and momentum transfer • Initiated at Saclay and perfected at NIKHEF, Amsterdam
• Also done at Mainz and currently at Jefferson Lab, VA
• Momentum dependence of cross section dominated by the corresponding sp wave function of the nucleon before it is removed
QMPT 540
Momentum profiles for nucleon removal • Closed-shell nuclei • NIKHEF data, L. Lapikás, Nucl. Phys. A553, 297c (1993)
QMPT 540
But...• Spectroscopic factors substantially smaller than simple IPM
QMPT 540
Remember• 208Pb sp levels
0f
QMPT 540
Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis
• S(2s1/2)=0.65
• other data:
• n(2s1/2)=0.75
• very different from atoms
QMPT 540
Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis (1988)
• start of strong fragmentation
• also very different from atoms
QMPT 540
Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis
• deeply bound states: strong fragmentation
• again different from atoms
QMPT 540
16O data from Saclay• Simple interpretation! • Mougey et al., Nucl. Phys. A335, 35 (1980)
Moment
um
Energy
QMPT 540
Recent Pb experiment• 100 MeV missing energy • 270 MeV/c missing momentum
• complete IPM domain
SRCalso LRC
QMPT 540
Reading• Read one of:
– Rev. Mod. Phys. 69, 981 (1997) --> nuclei (e,e’p)
– Rev. Mod. Phys. 67, 713 (1995) --> solids (e,2e)
QMPT 540
Sp propagator in many-body system• Similar definition as in sp problem • Also very useful both for discrete and continuum problems
• Fermion definition
• with normalized Heisenberg ground state
• Heisenberg picture operators
• and time-ordering operation is defined according to (fermions)
H |�N0 � = EN
0 |�N0 �
a�H (t) = ei� Hta�e�
i� Ht
a†�H(t) = e
i� Hta†�e�
i� Ht
T [a�H (t)a†⇥H(t�)] ⇥ �(t� t�)a�H (t)a†⇥H
(t�)� �(t� � t)a†⇥H(t�)a�H (t)
G(↵,�; t� t0) = � i
~ h N0 | T [a↵H
(t)a†�H(t0)] | N
0 i
QMPT 540
Lehmann representation• Introduce FT for practical applications
• Use again integral representation of step function
• Any sp basis can be used
• Still “wave functions” and eigenvalues as in sp problem!!
G(�,⇥;E) =� ⇤
�⇤d(t� t⇥) e
i� E(t�t�) G(�,⇥; t� t⇥)
G(�,⇥;E) =�
m
⇥�N0 | a� |�N+1
m ⇤ ⇥�N+1m | a†⇥ |�N
0 ⇤E � (EN+1
m � EN0 ) + i⇤
+�
n
⇥�N0 | a†⇥ |�N�1
n ⇤ ⇥�N�1n | a� |�N
0 ⇤E � (EN
0 � EN�1n ) � i⇤
= ⇥�N0 | a�
1E � (H � EN
0 ) + i⇤a†⇥ |�N
0 ⇤
+ ⇥�N0 | a†⇥
1E � (EN
0 � H) � i⇤a� |�N
0 ⇤
QMPT 540
Interaction picture• Split Hamiltonian • with problem solved (and corresponding time evolution)
• Define
• as the interaction picture state ket • Corresponding equation of motion
• where
• In general and do not commute!
H = H0 + H1
H0
|�I(t)� = exp�
i
�H0t
⇥|�S(t)�
i� �
�t|�I(t)⇥ = �H0 |�I(t)⇥+ exp
⇤i
�H0t
⌅i� �
�t|�S(t)⇥
= �H0 |�I(t)⇥+ exp⇤
i
�H0t
⌅ �H0 + H1
⇥|�S(t)⇥
= H1(t) |�I(t)⇥
H1(t) = exp�
i
�H0t
⇥H1 exp
�� i
�H0t
⇥
H0 H1
QMPT 540
Operators in the interaction picture• Consider in Schrödinger picture
• Go to interaction picture
• with
• is the corresponding operator in the interaction picture
OS |�S(t)� = |��S(t)�
|��I(t)⇥ = exp
�i
�H0t
⇥|��
S(t)⇥ = exp�
i
�H0t
⇥OS |�S(t)⇥
= exp�
i
�H0t
⇥OS exp
�� i
�H0t
⇥exp
�i
�H0t
⇥|�S(t)⇥
= OI(t) |�I(t)⇥
OI(t) = exp�
i
�H0t
⇥OS exp
�� i
�H0t
⇥
QMPT 540
Equation of motion in the interaction picture• Consider
• Example – in its own basis
– so
– and therefore and
i� ⇥
⇥tOI(t) =
�i� ⇥
⇥texp
�i
�H0t
⇥⇥OS exp
�� i
�H0t
⇥
+ exp�
i
�H0t
⇥OS
�i� ⇥
⇥texp
�� i
�H0t
⇥⇥
= �H0OI(t) + OI(t)H0
=⇤OI(t), H0
⌅
H0 =�
�
��a†�a�
i� ⇤
⇤ta�I (t) =
⇤a�I (t), H0
⌅
= exp�
i
�H0t
⇥ ⇤a�, H0
⌅exp
�� i
�H0t
⇥
= ��a�I (t)a†�I
(t) = ei⇥�t/�a†�a�I (t) = e�i⇥�t/�a�
QMPT 540
Components of Hamiltonian• Immediately
• and
• These operators have simple time dependence
• Critical operator: time-evolution in interaction picture
VI(t) =12
�
�⇥⇤⌅
(�⇥|V |⇤⌅) a†�I(t)a†⇥I
(t)a⌅I (t)a⇤I (t)
UI(t) =�
�⇥
(�|U |⇥) a†�I(t)a⇥I (t)
QMPT 540
Interaction picture time-evolution operator• Define • Note subscript “I” suppressed on evolution operator
• Obviously
• Explicit construction
• and therefore
|�I(t)⇥ = U(t, t0) |�I(t0)⇥
U(t0, t0) = 1
|�I(t)⇥ = exp�
i
�H0t
⇥|�S(t)⇥
= exp�
i
�H0t
⇥exp
�� i
�H(t� t0)⇥
|�S(t0)⇥
= exp�
i
�H0t
⇥exp
�� i
�H(t� t0)⇥
exp�� i
�H0t0
⇥|�I(t0)⇥
U(t, t0) = exp�
i
�H0t
⇥exp
�� i
�H(t� t0)⇥
exp�� i
�H0t0
⇥
QMPT 540
Some properties of evolution operator• Using previous result • Therefore unitary
• Note
• and • therefore
• For future applications combine SE in interaction picture with definition of evolution operator
so
• use boundary condition to integrate
U†(t, t0)U(t, t0) = U(t, t0)U†(t, t0) = 1
U†(t, t0) = U�1(t, t0)
U(t1, t2)U(t2, t3) = U(t1, t3)
U(t, t0)U(t0, t) = 1
U(t0, t) = U†(t, t0)
i� ⇥
⇥tU(t, t0) = H1(t)U(t, t0)
U(t, t0) = 1� i
�
� t
t0
dt� H1(t�)U(t�, t0)
i� �
�t|�I(t)� = H1(t) |�I(t)�
QMPT 540
Iterate• Use
• to generate expansion
U(t, t0) = 1� i
�
� t
t0
dt� H1(t�)U(t�, t0)
U(t, t0) = 1� i
�
⌃ t
t0
dt� H1(t�)
⇤1� i
�
⌃ t⇥
t0
dt�� H1(t��)U(t��, t0)
⌅
= 1 +��i
�
⇥ ⌃ t
t0
dt� H1(t�)
+��i
�
⇥2 ⌃ t
t0
dt�⌃ t⇥
t0
dt�� H1(t�)H1(t��) + ...
=⇥⇧
n=0
��i
�
⇥n ⌃ t
t0
dt1
⌃ t1
t0
dt2...
⌃ tn�1
t0
dtn H1(t1)H1(t2)...H1(tn)
QMPT 540
Example: second order
• introducing time-ordering
• Extend to all orders
• important for future applications
U(t, t0) =�⇤
n=0
��i
�
⇥n 1n!
⌅ t
t0
dt1
⌅ t
t0
dt2...
⌅ t
t0
dtn T⇧H1(t1)H1(t2)...H1(tn)
⌃
U2(t, t0) =�� i
�
⇥2 ⌥ t
t0
dt�⌥ t�
t0
dt�� H1(t�)H1(t��)
=12
�� i
�
⇥2⇧⌥ t
t0
dt�⌥ t�
t0
dt�� H1(t�)H1(t��) +⌥ t
t0
dt��⌥ t
t��dt� H1(t�)H1(t��)
⌃
=12
�� i
�
⇥2⇧⌥ t
t0
dt�⌥ t�
t0
dt�� H1(t�)H1(t��) +⌥ t
t0
dt�⌥ t
t�dt�� H1(t��)H1(t�)
⌃
=12
�� i
�
⇥2 ⇤⌥ t
t0
dt�⌥ t
t0
dt����(t� � t��)H1(t�)H1(t��) + �(t�� � t�)H1(t��)H1(t�)
⌅
=12
�� i
�
⇥2 ⌥ t
t0
dt�⌥ t
t0
dt�� T�H1(t�)H1(t��)
QMPT 540
Links with interaction picture• Use Schrödinger picture
• Note that
• and • For energy eigenkets
• so
• Also
OH(t) = exp�
i
�Ht
⇥OS exp
�� i
�Ht
⇥
= exp�
i
�Ht
⇥exp
�� i
�H0t
⇥OI(t) exp
�i
�H0t
⇥exp
�� i
�Ht
⇥
= U(0, t)OI(t)U(t, 0)
|�H� = |�S(t = 0)� = |�I(t = 0)�OS = OH(t = 0) = OI(t = 0)
|�nS (t)� = e�iEnt/� |�n�
= e�iHt/� |�n�|�n� = |�nH �
|�H⇥ = |�I(0)⇥ = U(0, t0) |�I(t0)⇥
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