Download - Pictures in Quantum Mechanics
![Page 1: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/1.jpg)
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)⇥
![Page 2: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/2.jpg)
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
⇥
![Page 3: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/3.jpg)
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
⇥
![Page 4: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/4.jpg)
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)�
![Page 5: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/5.jpg)
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 �
![Page 6: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/6.jpg)
�+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)
![Page 7: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/7.jpg)
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
![Page 8: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/8.jpg)
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 ⇥
![Page 9: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/9.jpg)
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 ⇥
![Page 10: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/10.jpg)
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)
⇥
⌅
![Page 11: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/11.jpg)
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
![Page 12: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/12.jpg)
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⌅
⇥
![Page 13: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/13.jpg)
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⇥
⇥
![Page 14: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/14.jpg)
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
![Page 15: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/15.jpg)
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)
![Page 16: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/16.jpg)
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)
![Page 17: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/17.jpg)
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)
![Page 18: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/18.jpg)
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
![Page 19: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/19.jpg)
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
![Page 20: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/20.jpg)
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�
![Page 21: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/21.jpg)
QMPT 540
Argon spectroscopic factors• s strength also in the continuum: Ar++ + e • note vertical scale
• red bars: 3s fragments exhibit substantial fragmentation
8%
![Page 22: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/22.jpg)
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
![Page 23: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/23.jpg)
QMPT 540
Momentum profiles for nucleon removal • Closed-shell nuclei • NIKHEF data, L. Lapikás, Nucl. Phys. A553, 297c (1993)
![Page 24: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/24.jpg)
QMPT 540
But...• Spectroscopic factors substantially smaller than simple IPM
![Page 25: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/25.jpg)
QMPT 540
Remember• 208Pb sp levels
0f
![Page 26: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/26.jpg)
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
![Page 27: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/27.jpg)
QMPT 540
Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis (1988)
• start of strong fragmentation
• also very different from atoms
![Page 28: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/28.jpg)
QMPT 540
Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis
• deeply bound states: strong fragmentation
• again different from atoms
![Page 29: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/29.jpg)
QMPT 540
16O data from Saclay• Simple interpretation! • Mougey et al., Nucl. Phys. A335, 35 (1980)
Moment
um
Energy
![Page 30: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/30.jpg)
QMPT 540
Recent Pb experiment• 100 MeV missing energy • 270 MeV/c missing momentum
• complete IPM domain
SRCalso LRC
![Page 31: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/31.jpg)
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)
![Page 32: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/32.jpg)
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
![Page 33: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/33.jpg)
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 ⇤
![Page 34: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/34.jpg)
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
![Page 35: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/35.jpg)
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
⇥
![Page 36: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/36.jpg)
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�
![Page 37: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/37.jpg)
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)
![Page 38: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/38.jpg)
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
⇥
![Page 39: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/39.jpg)
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)�
![Page 40: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/40.jpg)
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)
![Page 41: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/41.jpg)
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��)
![Page 42: Pictures in Quantum Mechanics](https://reader030.vdocuments.net/reader030/viewer/2022012517/6191ac263cfa7708c22172e6/html5/thumbnails/42.jpg)
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)⇥