quantum mechanica - quantum mechanica, 2012 …strat102/qm/week12.pdfgraph of the beam splitting,...

Quantum Mechanica

Peter van der StratenUniversiteit Utrecht

Peter van der Straten (Atom Optics) Quantum Mechanica December 4, 2012 1 / 12

Overzicht van het college

1 Stern-Gerlach experiment (1922)

2 Spin operatoren

3 Meting van Sz4 Meting van Sy5 Elektron in een magneetveld

6 Stern-Gerlach experiment (revisited)

7 Optellen van impulsmomenten

8 Spin drag (dreggen)


Stern-Gerlach experiment (1922)


Stern-Gerlach experiment (1922)

and classical views.”8

From Gedanken to DankenAfter hatching his idea in a warm bed, Stern hastened toBorn, but met a cool reception. In his autobiography, Bornsaid,

It took me quite a time before I took this ideaseriously. I thought always that [space] quan-tization was a kind of symbolic expression forsomething which you don’t understand. But totake this literally like Stern did, this was hisown idea. . . . I tried to persuade Stern thatthere was no sense [in it], but then he told methat it was worth a try.9

Happily, Stern found an eager recruit in Gerlach, who untilthen had not heard of space quantization.10

Despite Stern’s careful design and feasibility calcula-tions, the experiment took more than a year to accomplish.In the final form of the apparatus, a beam of silver atoms(produced by effusion of metallic vapor from an ovenheated to 1000°C) was collimated by two narrow slits (0.03mm wide) and traversed a deflecting magnet 3.5 cm longwith field strength about 0.1 tesla and gradient 10 tesla/cm. The splitting of the silver beam achieved wasonly 0.2 mm. Accordingly, misalignments of collimatingslits or the magnet by more than 0.01 mm were enough tospoil an experimental run. The attainable operating timewas usually only a few hours between breakdowns of theapparatus. Thus, only a meager film of silver atoms, toothin to be visible to an unaided eye, was deposited on thecollector plate. Stern described an early episode:

After venting to release the vacuum, Gerlachremoved the detector flange. But he could seeno trace of the silver atom beam and handedthe flange to me. With Gerlach looking overmy shoulder as I peered closely at the plate,we were surprised to see gradually emerge thetrace of the beam. . . . Finally we realizedwhat [had happened]. I was then the equiva-lent of an assistant professor. My salary wastoo low to afford good cigars, so I smoked badcigars. These had a lot of sulfur in them, so my

breath on the plateturned the silver intosilver sulfide, which isjet black, so easily vis-ible. It was like devel-oping a photographicfilm.7

After that episode, Gerlachand Stern began using a pho-

tographic development process, although both continuedpuffing cigars in the lab. Still, recalcitrant difficulties per-sisted. As inconclusive efforts continued for months,Stern’s assessment of space quantization wavered betweenconviction and rejection. Gerlach also encountered doubt-ful colleagues, including Debye, who said, “But surely youdon’t believe that the [spatial] orientation of atoms issomething physically real; that is [only] a timetable for theelectrons.”10

Another handicap was the financial disarray thatbegan to beset Germany. Born was unstinting in efforts toraise funds to support the SGE. He took advantage of thegreat interest in Einstein and relativity theory by pre-senting a series of public lectures “in the biggest lecture-hall of the University . . . and charged an entrance fee. . . .The money thus earned helped us for some months, but asinflation got worse . . . new means had to be found.”8 Bornmentioned this situation “jokingly” to a friend who was de-parting on a trip to New York; he was incredulous when,a few weeks later, a postcard arrived simply saying thathe should write to Henry Goldman and giving the address:

At first I took it for another joke, but on re-flection I decided that an attempt should bemade. . . . [A] nice letter was composed and dis-patched, and soon a most charming reply ar-rived and a cheque for some hundreds of dol-lars. . . . After Goldman’s cheque had saved ourexperiments, the work [on the Stern–Gerlachexperiment] went on successfully.9

Goldman, a founder of the investment firm Goldman Sachsand progenitor of Woolworth Co stores, had family roots inFrankfurt.

Meanwhile, Stern had moved to the University of Ro-stock as a professor of theoretical physics. In early 1922,he and Gerlach met in Göttingen to review the situationand decided to give up. However, a railroad strike delayedGerlach’s return to Frankfurt, giving him a long day to goover all the details again. He decided to continue, im-proved the alignment, and soon achieved a clear splittinginto two beams.5 Stern recalled that his own surprise andexcitement were overwhelming when he received atelegram from Gerlach with the terse message: “Bohr isright after all.”11 Gerlach also sent a postcard to Bohr with

56 December 2003 Physics Today http://www.physicstoday.org

Figure 4. Gerlach’s postcard,dated 8 February 1922, toNiels Bohr. It shows a photo-graph of the beam splitting,with the message, in transla-tion: “Attached [is] the exper-imental proof of directionalquantization. We congratu-late [you] on the confirma-tion of your theory.” (Cour-tesy AIP Emilio Segrè VisualArchives.)


Stern-Gerlach experiment


Spin drag

Equal mixture spin up and down


Spin drag

F↑

Apply force on spin up


Spin drag

F↑

No drag - acceleration


Spin drag

F↑

F↑

Drag - constant velocity difference


Condensed matter vs. cold atoms

Condensed matter Cold atoms

electrons atomscharge masselectric field magnetic field gradientelectron spin pseudo-spin 1/2e-e interactions atom-atom interactionse-photon interactions -e-impurities interactions -current absorption imaging


Spin drag

Js

Jm

y

x

Js

Charge or mass current conserved

Spin current switched direction

Strong damping


Measurement – Constant force

0.0

0.3

0.6

0.9

1.2

1.5

[mm

]

0 1 2 3 4 5 6[mm]

0 1 2 3 4 5 6[mm]

0.2

0.4

0.6

0.8

1.0

1.2

Tra

nsm

issi

on


Drag rate – Constant force

0 20 40 60 80 100t[ms]

−0.4

−0.3

−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

∆x

[mm

]

(Coll. rate: 137± 18 Hz)

Drag rate: 88± 6 Hz


Drag rate vs. Collision rate (Classical)

0 50 100 150 200γ↑↓ [Hz]

0

50

100

150

200γ

[Hz]

Ratio γ/γ↑↓ = 2/3 independent of temperatureL. Vichi, S. Stringari, Phys. Rev. A 60, 4734 (1999)


Drag rate vs. Fugacity (Quantum regime)

0.0 0.2 0.4 0.6 0.8 1.0Exp(µ/(kBT ))

0

10

20

30

40

50

60

70

80

90

γ/T

2[H

z/µ

K2]

Bose enhancement → precursor of the phase transition

Predicted by ab initio theory - no fitted parameters

H.J. van Driel, R.A. Duine, & H.T.C. Stoof, Phys. Rev. Lett. 105, 155301 (2010)



0.0 0.2 0.4 0.6 0.8 1.0Exp(µ/(kBT ))

0

10

20

30

40

50

60

70

80

90

γ/T

2[H

z/µ

K2]

Bose enhancement

→ precursor of the phase transition





0.0 0.2 0.4 0.6 0.8 1.0Exp(µ/(kBT ))

0

10

20

30

40

50

60

70

80

90

γ/T

2[H

z/µ

K2]

Bose enhancement → precursor of the phase transition




quantum mechanica - quantum mechanica, 2012 …strat102/qm/week12.pdfgraph of the beam splitting,...

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