Wehavearelaonshipbetweenk2andthe energye k e 2i 2 2

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Unformatted text preview: if
the
poten,al
energy
changes
are
 finite.
 lim ψ ( x + ε ) = ψ ( x ) ε →0 dψ ( x + ε ) dψ ( x ) lim = ε →0 dx dx For
“Par,cle
in
a
square
well”
the
changes
in
poten,al
are
NOT
finite.
 Par,cle
in
a
ring
 (in
the
spirit
of
the
Bohr’s
model)
 •  Bohr
made
an
ad‐hoc
assump,on
that
the
 electron
must
use
well
defined
angular
 momentum
values.

 J = mvr = n n = 1, 2, 3, ... h = 2π h : Planck constant a
 Can
we
obtain
this
result
in
a
 more
natural
way?
 What
is
the
wave
func,on
of
a
par,cle
 in
a
ring
with
radius
“a”?
 •  The
wave
func,on
and
its
first
deriva,ve
must
be
 con,nuous
 •  We
can
also
require
that
the
wave
func,on
must
 be
periodic
in
the
angle
ϕ:
 –  
ψ(ϕ+2π)=ψ(ϕ)
 But
we
s,ll
need
an
 equa,on
to
determine
 Ψ(ϕ)
 ϕ
 The
Schroedinger
Equa,on
for
a
 par,cle
in
a
ring
 2 d 2 − ψ (φ ) = Eψ (φ ) 2 2 2 ma dφ ψ : Wave function / Eigenfunction φ...
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