Lecture 33: Quantum Mechanical Spin

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Job Johnston
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1 Lctu 33: Quantu Mcancal pn Py85 Fall 9

2 Intnc pn Epcally w av foun tat ot patcl av an atonal ntnal g of fo call pn T tn-glac pnt 9): Eac typ of patcl a a ct nub of ntnal tat: tat --> pn _ 3 tat --> pn Etc.

3 Intptaton It bt to tnk of pn a jut an atonal quantu nub n to pcfy t tat of a patcl. Wtn t Dac foal t latvly pl an qu no nw pycal concpt T pycal anng of pn not wlluntoo Fo Dac q. w fn tat fo QM to b Lont nvaant qu patcl to av bot ant-patcl an pn. T pn of a patcl a fo of angula ontu

4 pn Opato pn cb by a vcto opato: T coponnt atfy angula ontu coutaton laton: T an ultanou gntat of an t: y y y y y ] [ ] [ ] [ y )

5 Allow quantu nub Fo any t of 3 opato atfyng t angula ontu algba t allow valu of t quantu nub a: j { 3 K} { j j K j} j Fo obtal angula ontu t allow valu w fut tct to only ntg valu by t qunt tat t wavfuncton b ngl-valu Fo pn t quantu nub can only tak on on valu T valu pn on t typ of patcl : Hgg /: Elcton poton poton nuton uonnutno quak : Poton W Z Gluon : gavton { K }

by t qunt tat t wavfuncton b ngl-valu Fo pn t quantu nub can only tak on on valu T valu pn

6 Coplt ngl patcl ba A t of 5 coutng opato wc cb t npnnt obvabl of a ngl patcl a: R O quvalntly: R L L o pobl ba coc: { } { p } { l } l { n l } l Wn alng wt a ngl-patcl t pbl to op t quantu nub

7 Intnc Magntc Dpol Mont Du to pn an lcton a an ntnc agntc pol ont: g g t lcton g-facto Fo an lcton w av: T t ot pcly au pycal quantty Fo ot pupo w can tak g o tat g ±.5 Fo any cag patcl w av: g q M Eac patcl a a ffnt g-facto

8 Haltonan fo an lcton n a agntc fl cau t lcton a pont-patcl t pol-appoaton alway val fo t pn g of fo Any `kntc ngy aocat wt abob nto t t a To obtan t full Haltonan of an lcton w ut a a ngl t: H H R) H [ P A R) ] Ö R) R)

9 Unfo Wak Magntc Fl a a ptubaton Fo a wak unfo fl w fn P H L ) Wt t aton of a pcally ytc potntal t gv: P H V R) L ) If t o-fl gntat a known H n n l n l l E T wak-unfo-fl gntat a: { n l } l H n l l En n l l n E n l l ) E l o Magnton l

10 Wavfuncton In Dac notaton all pn o a two ta quantu nub T paat concpt of a pno unncay Coonat ba: Egntat of { } R Pojcto: I V 3 Wavfuncton: :

11 pno Notaton: W tnk of t a coponnt of a lngt vcto w ac coponnt a wavfuncton Eapl: / : : : pno wavfuncton fnton: : [ ] If tnal an ntnal oton a not ntangl w can facto t pno wavfuncton: c c [ ] : c c Not tat cnng o unuual I tn a pu pno

12 cöng' Equaton W tat fo: Ht fo lft wt wt Int t pojcto t Lt: Fo /: [ ] H t H t H V P M 3 H [ I ] [ V ] R) V ) I [ ] V ) V ) V ) V ) t M [ ] [ ] [ V] ][ ]

13 Eapl: Elcton n a Unfo Fl [ ] [ ] [ ] [ ][ ] V M t t φ t t φ φ T jut a pntaton of two paat quaton: W woul av av at t a quaton ung Dac notaton wtout v ntonng pno

14 Paul Matc H w tat w av cov on of t Paul Matc: T ot Paul atc a: Tn n t ba of gntat of w av: If w only ca about pn ynac: t φ σ σ σ y σ [ ] [ ] M t σ φ [ ] [ ] c c t σ

15 Two patcl wt pn How o w tat a yt of two patcl wt a M an M cag q an q an pn an? a: Wavfuncton: Haltonan w/out otonal g of fo: Haltonan w/ otonal g of fo: ; ; ) : ) ) R M q R M q H ) ) ) ) ) ) R M q R q R A q P M R M q R q R A q P M H Φ Φ

16 Eapl # A pn _ patcl n t tat wt pct to t -a. Wat t pobablty of fnng t n t -tat wt pct to t -a? Lt: { } In t ba t opato fo t -coponnt of pn : σ y yty σ ut av gnvalu an - T gnvcto coponng to - fn by: σ

17 Eapl # contnu: T pl tat: σ σ σ σ P

18 Eapl # Two ntcal pn-/ patcl a plac n a unfo agntc fl. Ignong otonal g of fo wat a t ngy-lvl an gnac of t yt? tat: { } Z-a con along -fl Haltonan: gq M ) H a tat a alay gntat: H gq M H H H gq M gq E ; M E ; gq E3 ; 3 M

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