Lecture 3: The Time Dependent Schr

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Lecture 3 The Time Dependent Schrödinger
Equation The material in this lecture is not
covered in Atkins. It is required to understand
postulate 6 and 11.5 The informtion of a
wavefunction Lecture on-line The Time
Dependent Schrödinger Equation (PDF) The time
Dependent Schroedinger Equation (HTML) The time
dependent Schrödinger Equation (PowerPoint)
Tutorials on-line The postulates
of quantum mechanics (This is the writeup for
Dry-lab-II ( This lecture coveres parts of
postulate 6) Time
Dependent Schrödinger Equation The
Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audio-visuals on-line review of
the Schrödinger equation and the Born postulate
(PDF) review of the
Schrödinger equation and the Born postulate
(HTML) review of
Schrödinger equation and Born postulate
(PowerPoint , 1MB) Slides from
the text book (From the CD included in Atkins
,)
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It is important to note that the particle is not
distributed over a large region as a charge
cloud
It is the probability patterns (wave function)
used to describe the electron motion that behaves
like waves and satisfies a wave equation
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We have
and
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The R.H.S. does not depend on t if we now assume
that V is time independent. Thus, the L.H.S.
must also be independent of t
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Thus
The L.H.S. does not depend on x so the R.H.S.
must also be independent of x and equal to the
same constant, E.
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We can now solve for f(t)
Or
Now integrating from time t0 to tto on both
sides affords
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Or
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This is the time-independent Schroedinger
Equation for a particle moving in the time
independent potential V(x)
It is a postulate of Quantum Mechanics that E is
the total energy of the system
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The total wavefunction for a one-dimentional
particle in a potential V(x) is given by
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Thus we can write without loss of generality for
a particle in a time-independent potential
This wavefunction is time dependent and complex.
Let us now look at the corresponding probability
density
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Thus , states describing systems with a
time-independent potential V(x) have a
time-independent (stationary) probability
density.
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We say that systems that can be described by wave
functions of the type
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