COORDINATE TRANSFORMATIONS:
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H WAVEFUNCTIONS AND PROBABILITIES:
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EXPECTATION VALUES:
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EXPECTATION VALUES
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TYPES OF EXPECTATION VALUES:
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EXPECTATION VALUES – EXAMPLES:
We will consider several examples where
expectation values can be calculated for
energy, linear momentum, position and
distance of the electron from the nucleus.
Additional calculations are possible – for
example, expectation values for various
types of angular momentum. Initially, the
form of the wave function is known.
PIAB – ONE DIMENSION – MOMENTUM:
The result obtained for the linear
momentum expectation value on the
previous slide, zero, may seem strange. In
the one dimensional PIAB the particle always
has non-zero momentum and kinetic energy.
However, the average or expectation value
for momentum is zero since the particle
moves (equal probability) in both directions.
EXPECTATION VALUE – POSITION:
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EXPECTATION VALUES – HYDROGEN ATOM:
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VARIATIONAL CALCULATIONS:
In all examples of expectation value
calculations considered the exact form of the
wave function was known. In some cases the
precise form of the wave function is not
known. We can still estimate values for
energies and so on using the methods
outlined and an approximate, guessed or
trial wave function.
VARIATIONAL CALCULATIONS:
Estimates for energies obtained using
approximate wave functions are extremely
important in chemistry. If the energy
calculated using the true wave function is
ψExact then with and approximate wave
function, ψAPP, we would obtain an
approximate estimate for energy, EAPPROX.
VARIATIONAL CALCULATIONS:

VARIATIONAL THEOREM:
The variational theorem tells us that when
any approximate wave function is used the
estimate for energy obtained from an
expectation value calculation will be greater
than the energy value calculated with the
true wave function. We will explore this
using class and assignment examples.