The Bohr model is discussed in more detail
elsewhere in FLAP.
Niels Bohr (1885–1962)
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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S570 V1.1
You will often find h/2π written as ˙.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
COPYRIGHT © 1998
THE OPEN UNIVERSITY
S570 V1.1
Louis de Broglie (1892–1987).
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
The wave equation for classical waves is
introduced elsewhere in FLAP, as is the
Schrödinger equation for de Broglie
waves.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
Quantum mechanical operators and
eigenvalue equations are discussed more
fully elsewhere in FLAP.
See the Glossary for details.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
Operators are distinguished by the use of
Roman (upright) characters with the carat
symbol ^ (or hat) written above the
quantity.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
Partial differentiation is introduced
elsewhere in FLAP. When evaluating a
partial derivative such as ∂0 /∂x the basic
rule is to treat all variables other than x as
though they are constants.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
We will from now on use E instead of E tot;
it is to be understood that it is always the
total energy that we are interested in.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
Strictly speaking we are here describing
the orbital magnetic quantum number,
which we designated m0l to distinguish it
from the spin magnetic quantum number
m0s associated with the intrinsic magnetism
of the electron itself. In this module we are
not concerned with electron spin and so
there is no ambiguity in simply using m
here, as in Yl1 m0(θ, φ ).
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
There is nothing special about the z-axis
itself; it is just that the angular coordinate
system chosen singles out the z-axis as the
one from which the angle θ is measured.
The point is that the magnitude of the
angular momentum and its component in
one direction can be known
simultaneously. We decide to call this
direction the z-direction.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
COPYRIGHT © 1998
THE OPEN UNIVERSITY
S570 V1.1
Remember: if z = eiα with α real, then
|1z1|2 = zz* = eiα1e−iα = 1.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
COPYRIGHT © 1998
THE OPEN UNIVERSITY
S570 V1.1
It should be noted that there are an infinite
number of possible bound states.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
Pieter Zeeman (1865–1943).
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
The best way to formulate the
correspondence principle has been a matter
of debate for many decades. Our
formulation is not meant to be definitive.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
✧
Remembering that a0 = 0.53 × 10−101m,
E1 = −13.61eV, and calculating that
v 1 = 2.18 × 1061m1s−1, we find
r n = 10 12 × 0.53 × 10 −101m = 531m
(a very large hydrogen atom!)
v n = 2.181m1s−1
∆r/r = 2 × 10 −6, or ∆r ≈ 10−41m
∆E ≈ 2.7 × 10−171eV
∆v ≈ −2 × 10−61m1s−1
so that when the change in orbital radius is
just approaching visibility, the changes in
energy and electron speed are negligibly
small.
FLAP P11.3
Schrödinger’s model of the hydrogen atom
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THE OPEN UNIVERSITY
S570 V1.1
In fact, we can treat all these quantities as
varying continuously for sufficiently large
n. This situation would be
indistinguishable from classical
predictions.4❏
FLAP P11.3
Schrödinger’s model of the hydrogen atom
COPYRIGHT © 1998
THE OPEN UNIVERSITY
S570 V1.1