Chapter 2
Sketch
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Sketch
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Sketch
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From Problem , phase ?2?x???t
= constant Then
2???dxdt???0,?dxdt?????p?????2??? From Problem , phase ?2?x???t
= constant Then
2???dxdt???0,?dxdt????????p??2??? _______________________________________
E?h??hc???hc?E
Gold: E?4.90eV??4.90??1.6?10?19?
J So,
?6.625?10?34??3?1010?????5?4.90??1.6?10?19?2.54?10cm
or
??0.254?m
Cesium: E?1.90eV??1.90??1.6?10?19?J
So,
?3410???6.625?10??3?10??5?1.90??1.6?10?19??6.54?10cm
or
??0.654?m
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h6.625?10?34 (a) p???550?10?9
?1.205?10?27kg-m/s
??p1.2045?10?273m?9.11?10?31?1.32?10m/s or ??1.32?105cm/s
p?h6.625?10?34 (b) ??440?10?9
?1.506?10?27kg-m/s
p1.5057?10?27????1.65?103m9.11?10?31m/s or ??1.65?105cm/s (c) Yes
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(a) (i)
p?2mE?2?9.11?10?31??1.2??1.6?10?19?
?5.915?10?25kg-m/sh6.625?10?34 ??p?5.915?10?25?1.12?10?9m or ??11.2Ao
(ii)p?2?9.11?10?31??12??1.6?10?19? ?1.87?10?24kg-m/s ?34 ??6.625?101.8704?10?3.54?10?10?24m or ??3.54Ao
(iii) p?2?9.11?10?31??120??1.6?10?19? ?5.915?10?24kg-m/s ?34 ??6.625?105.915?10?24?1.12?10?10m or ??1.12Ao
(b)
p?2?1.67?10?27??1.2??1.6?10?19? ?2.532?10?23kg-m/s??6.625?10?34 2.532?1023?2.62?10?11?m or ??0.262Ao
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Eavg?3?2kT??3??2???0.0259??0.03885eV Now
pavg?2mEavg
? ?
?29.11?10?31?0.03885??1.6?10?19?
or
pavg?1.064?10?25kg-m/s Now
??h6.625?10?34p?1.064?10?25?6.225?10?9m or
??62.25Ao
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Ep?h?hcp??
p
Now
Ep2ee?2m and
ph1?h?2e???Ee???e2m???e?? Set Ep?Ee and ?p?10?e Then
22 hc1??h??1??????10h??p2m???e?2m???p? ? which yields
?100hp?2mc Ehchc2mc2 p?E????2mc? p100h100 ?2?9.11?10?31??3?108?2100
?1.64?10?15J ?10.25keV _______________________________________
p?h6.625?10?34(a)??85?10?10
?7.794?10?26 kg-m/s
??p7.m?794?10?2649.11?10?31?8.56?10m/s or ??8.56?106cm/s
E?1m?2?12?9.11?10?31??8.56?1042?2
?3.33?10?21J
or E?3.334?10?211.6?10?19?2.08?10?2eV (b) E?1?9.11?10?31??8?103?22 ?2.915?10?23J
or E?2.915?10?23?1.82?10?41.6?10?19eV p?m???9.11?10?31??8?103? ?7.288?10?27kg-m/s
h6.625?10?35??p?7.288?10?27?9.09?10?8m or ??909Ao
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(a) E?h??hc?6.625?10?348????3?10?1?10?10 ?1.99?10?15J
Now
E?e?V?V?E1.99?10?15 e?1.6?10?19
V?1.24?104V?12.4kV
(b)
p?2mE?2?9.11?10?31??1.99?10?15?
?6.02?10?23kg-m/s
Then
h6.625?10?34??p?6.02?10?23?1.10?10?11m or
??0.11Ao
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?p??1.054?10?34 ?x?10?6 ?1.054?10?28kg-m/s
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(a) (i) ?p?x??
?p?1.054?10?34?2612?10?10?8.783?10kg-m/s (ii)?E?dEd?p2?dp??p?dp?????p ?2m?? ?2p2m??p?p?pm Now p?2mE
?2?9?10?31??16??1.6?10?19? ?2.147?10?24kg-m/s
?2.1466?10?24??8.?26so ?E?783?10?9?10?31
?2.095?10?19J
or ?E?2.095?10?191.6?10?19?1.31eV
(b) (i) ?p?8.783?10?26kg-m/s (ii)p?2?5?10?28??16??1.6?10?19? ?5.06?10?23kg-m/s
?5?23?26 ?E?.06?10??8.783?10?5?10?28
?8.888?10?21J
?E?8.888?10?21 or 1.6?10?19?5.55?10?2eV _______________________________________
?p???x?1.054?10?3410?2?1.054?10?32kg-m/s
p?m??????p1.054?10?32m?1500 ???7?10?36m/s
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(a) ?E?t??
?t?1.054?10?34?0.8??1.6?10?19??8.23?10?16s
??1.054?10?34(b) ?p?x?1.5?10?10 ?7.03?10?25kg-m/s
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(a) If ?1?x,t? and ?2?x,t? are
solutions to
Schrodinger's wave equation, then
??22??1?x,t?2m???1?x,t??x2?V?x??1?x,t??j??t and
??222m???2?x,t??x2?V?x??2?x,t??j???2?x,t??t Adding the two equations, we obtain
??2?22m??x2??1?x,t???2?x,t?? ? ?Vx???1?x,t???2?x,t??
?j???t??1?x,t???2?x,t??
which is Schrodinger's wave equation.
So
?1?x,t???2?x,t? is also a solution.
(b) If ?1?x,t???2?x,t? were a
solution to
Schrodinger's wave equation, then we could write
??2?2 2m??x2??1??2??V?x???1??2? ?j???t??1??2?
which can be written as
??2
?2m???2?2?2?1??1??2??1?x2??2?x2?2?x??x? ?
?V?x???????2??1?1??2??j???1?t??2?t?? Dividing by ?1??2, we find
??2?2m?1??2?21?2?12??1??2??2?2?x??2???x?x?1?x?1?2? ?V?x??j???1??2???1??1?2?t?1?t??
Since ?1 is a solution, then ??22m?1???2?1?x2?V?x??j??1??1?? 11?t Subtracting these last two equations,
we have
??2 ?2m?1?2?22??1??2???2??2?x?1?2?x?x? ?
?j??1??2?? 2?t Since ?2 is also a solution, we have
2 ??22m?1????2?V?x??j??1??2?? 2?x22?t Subtracting these last two equations,
we obtain
??2????2m?2??1?2?V?x??0
1?2?x?x This equation is not necessarily valid, which
means that ?1?2 is, in general, not a solution
to Schrodinger's wave equation. _______________________________________
?3
?A2cos2???x??dx?1?1?2?
A2??xsin??x???3?2?2????1 ?1 A2??3?2????1??2??????1
so A2?12
or A?12
半导体物理与器件第四版课后习题答案2
