WEBVTT
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Lecture 4,
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a review of mathematics
for quantum chemistry.
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In this lecture,
we quickly review some
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of the basic
mathematical concepts
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that are essential
for quantum mechanics.
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If you are not already familiar
with the topics of this lecture,
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I urge you to take advantage
of the office hours
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of the instructor
and teaching assistants
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to discuss ways to catch up.
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Specifically, we will refresh
our existing knowledge
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of complex numbers,
matrix algebra,
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and partial derivatives,
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in relation
to quantum mechanics.
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We live
in the 3-dimensional universe
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or 4-dimensional spacetime.
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Our life would be very difficult
if one or more
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of these dimensions
were suppressed.
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For example, if the z-axis
were not available,
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we would have to drive
every day from home
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to the university campus,
instead of just fly in.
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Numbers are known to exist
in 2-dimensional space
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known as the complex plane.
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Describing physics and chemistry
would be extremely difficult
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or impossible
if the imaginary axis
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of the complex plane
were not available.
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A most general number
is a complex number,
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and it consists
of the real part a
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and the imaginary part b
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* the imaginary unit i,
which is √-1.
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Here, a and b are real numbers.
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So, we can associate any complex
number a+bi to a point
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in the 2-dimensional
complex plane,
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whose x-axis value is a
and y-axis value is b.
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Pure real numbers exist
on the x-axis,
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which is called the real axis.
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Pure imaginary numbers exist
on the y-axis,
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which is called
the imaginary axis.
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All conceivable numbers
are confined
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in this 2-dimensional plane.
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Every attempt to generate
a new kind of number
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that is outside this
2-dimensional plane has failed.
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You will be asked to try this,
as homework.
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These are 3 important
attributes of a complex number.
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The complex conjugate of complex
number z is designated by z*
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and its value is obtained
by replacing all i’s by -i’s
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wherever they appear.
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In the complex plane,
the complex conjugate operation
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brings z
to its mirror image position
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with respect to the x-axis.
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The absolute value
of complex number z is √(z*z)
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and is calculated
to be √(a^2+b^2).
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It is always real
and nonnegative,
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and is the distance between
the point representing z
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and the origin
in the complex plane.
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The argument or phase
of z is the angle
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between the vector
from the origin to the point z
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and the positive
real-axis vector.
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We adopt this 2-dimensional
geometrical view of numbers.
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So the standard form
of any complex number
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is r(cos(θ)+ i sin(θ)),
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where r is the absolute value
of the complex number
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and θ is its argument
or phase angle.
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Since both cosine and sine
functions are sinusoidal waves
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of phase angle θ,
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we may say that
a complex number is a wave
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and well suited to describe
quantum mechanics involving
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wave-particle duality,
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as we have seen
in the previous lecture.
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Suppose we have two
complex numbers z_1 and z_2,
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which are expressed
in the standard form like so.
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Now let r_1 and r_2
be equal to unity
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to simplify our discussion.
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Then, both z_1 and z_2 exist
somewhere on the circumference
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of the circle with the unit
radius centered at the origin.
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They have the phase angles
of θ_1 and θ_2.
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What would be z_1* z_2?
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If we perform the multiplication
using the standard forms,
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after using the trigonometric
identities we all hate,
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we obtain the product back
in the standard form.
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So, product z_1*z_2 is also
on the same circle
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with the unit radius
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with the phase angle
of θ_1+θ_2.
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The multiplication of two
complex numbers results
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in the addition
of their phase angles.
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The standard form can
be further compressed
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into a single exponential
with an imaginary exponent.
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This is the celebrated
Euler’s formula,
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which we have already seen.
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As homework, you will be asked
to prove this formula
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in two different ways.
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With this expression, we can
more immediately reach
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the same conclusion
in the previous page,
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bypassing the trigonometric
identities.
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The product
of two complex numbers
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with the unit absolute value
and phase angles of θ_1 and θ_2
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is the complex number
with the unit absolute value
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with the sum
of the phase angles, θ_1+θ_2.
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Also, we can easily see that
the raising a complex number
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to the power n results
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in multiplying the phase angle
by n.
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We can appreciate
the power of Euler’s form
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by the following example.
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This 28th order polynomial
equation should have
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28 distinct
complex-number roots.
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Can we solve for all roots?
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Yes, very easily if we express
the roots in Euler’s formula.
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Remember that to raise
a complex number
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with the unit absolute value
to the 28th power
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is to move its point
counterclockwise
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28 times its phase angle.
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So the roots are
on the circumference
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of the unit-radius circle
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and their phase angles should
evenly divide
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an integer multiple of 2π/28.
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So the distribution of all roots
in the complex plane would look
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like a pizza of the unit radius
divided evenly for 28 people.
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The matrix is everywhere
in computational sciences
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and engineering.
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This is because a computer is
extremely good at performing
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simple arithmetic operations
on a huge array of numbers.
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So, in many areas of science
and engineering
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with strong theoretical
underpinnings,
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we often translate
the theoretical formulations
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into matrix algebra
and lend them
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to quantitative
computer predictions.
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Moreover, quantum mechanics
has even stronger ties
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with matrix algebra.
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One of the two
initial formulations
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of quantum mechanics was made
in the language of matrices
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and was called matrix mechanics
of Heisenberg.
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It is equivalent
to the wave mechanics
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of Schrodinger,
which we learn in this course.
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I trust that you
already know how
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to multiply two square matrices.
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If not, please be sure to meet
with an instructor
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or a teaching assistant.
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It is important to remember
that matrix multiplication
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is not commutative,
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and so the products are
not the same depending
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on the order of multiplication.
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The identity or unit matrix
is a diagonal matrix,
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whose diagonal elements
are equal to 1.
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The multiplications
are commutative
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if one of the matrices
is the unit matrix.
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In fact, multiplication
by the unit matrix
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does not change the matrix
being multiplied by it.
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If a matrix is multiplied
by its inverse from the left
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or from the right,
the product is the unit matrix.
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A matrix may or may not
have an inverse.
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For example, this matrix
enclosed by blue curve
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is the inverse
of this abcd matrix.
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This inverse exists if this
denominator ad-bc is not 0.
00:07:48.968 --> 00:07:51.371 align:middle position:47% size:63%
And this denominator
is called the determinant
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of the abcd matrix.
00:07:53.606 --> 00:07:56.910 align:middle position:47% size:63%
So, in general,
the matrix inverse exists
00:07:56.910 --> 00:07:59.979 align:middle position:47% size:63%
if and only if
its determinant is not 0.
00:07:59.979 --> 00:08:04.884 align:middle position:44% size:78%
It is not easy to calculate the
determinant of a larger matrix.
00:08:04.884 --> 00:08:07.587 align:middle position:47% size:63%
Many important problems
of physics and chemistry,
00:08:07.587 --> 00:08:09.589 align:middle position:47% size:63%
including the Schrodinger
equation,
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can be cast into a matrix
eigenvalue problem.
00:08:14.027 --> 00:08:15.328 align:middle position:50% size:45%
In such a problem,
00:08:15.328 --> 00:08:18.498 align:middle position:50% size:70%
we search
for a nontrivial vector, xy,
00:08:18.498 --> 00:08:22.202 align:middle position:46% size:68%
and a complex value, e,
that satisfy this equation.
00:08:24.070 --> 00:08:27.674 align:middle position:50% size:61%
So, only the matrix abcd
is known,
00:08:27.674 --> 00:08:30.744 align:middle position:47% size:63%
and everything else is
unknown in this equation.
00:08:30.744 --> 00:08:33.480 align:middle position:47% size:58%
Clearly, zero vector
satisfies this equation
00:08:33.480 --> 00:08:36.015 align:middle position:45% size:73%
for any arbitrary value of e,
00:08:36.015 --> 00:08:39.285 align:middle position:50% size:70%
but we call such zero vector
solutions trivial solutions
00:08:39.285 --> 00:08:41.388 align:middle position:45% size:73%
and are uninterested in them.
00:08:41.388 --> 00:08:46.292 align:middle position:44% size:78%
We call the nontrivial solution
xy an eigenvector
00:08:46.292 --> 00:08:49.963 align:middle position:46% size:68%
and its corresponding value
of e an eigenvalue.
00:08:49.963 --> 00:08:53.466 align:middle position:50% size:50%
For an n-by-n matrix
eigenvalue equation,
00:08:53.466 --> 00:08:57.537 align:middle position:45% size:73%
n distinct nontrivial
solutions are known to exist.
00:08:57.537 --> 00:09:01.708 align:middle position:44% size:78%
So, in this 2-by-2 matrix case,
we have 2 eigenvalues
00:09:01.708 --> 00:09:03.910 align:middle position:48% size:48%
and 2 corresponding
eigenvectors.
00:09:05.345 --> 00:09:08.114 align:middle position:50% size:75%
We can stack
these two eigenvalue equations
00:09:08.114 --> 00:09:11.518 align:middle position:45% size:73%
side-by-side and column-wise.
00:09:11.518 --> 00:09:15.955 align:middle position:46% size:68%
We then obtain this single
matrix eigenvalue equation.
00:09:15.955 --> 00:09:20.560 align:middle position:45% size:73%
The two eigenvalues e and f
are now the diagonal elements
00:09:20.560 --> 00:09:23.863 align:middle position:50% size:65%
of the diagonal eigenvalue
matrix.
00:09:23.863 --> 00:09:26.399 align:middle position:50% size:50%
The two eigenvectors
are now two columns
00:09:26.399 --> 00:09:28.935 align:middle position:50% size:65%
of the eigenvector matrix.
00:09:28.935 --> 00:09:32.772 align:middle position:50% size:70%
Again, only the abcd matrix
is known
00:09:32.772 --> 00:09:34.974 align:middle position:50% size:75%
and everything else is unknown
00:09:34.974 --> 00:09:37.844 align:middle position:47% size:63%
except we also know that
the off-diagonal elements
00:09:37.844 --> 00:09:40.847 align:middle position:50% size:61%
of the eigenvalue matrix
are all 0.
00:09:40.847 --> 00:09:44.017 align:middle position:44% size:78%
I place the diagonal eigenvalue
matrix to the right
00:09:44.017 --> 00:09:46.252 align:middle position:50% size:65%
of the eigenvector matrix.
00:09:46.252 --> 00:09:49.389 align:middle position:50% size:81%
If I did the other way around,
the equation would be incorrect.
00:09:50.590 --> 00:09:52.592 align:middle position:48% size:48%
Please verify this.
00:09:52.592 --> 00:09:55.495 align:middle position:50% size:65%
If we multiply the inverse
of the eigenvector matrix
00:09:55.495 --> 00:09:57.897 align:middle position:44% size:78%
with both sides of the equation
from the left,
00:09:57.897 --> 00:09:59.499 align:middle position:50% size:61%
we obtain this equation.
00:10:01.334 --> 00:10:06.005 align:middle position:50% size:75%
So, solving this equation is
equivalent to finding a matrix
00:10:06.005 --> 00:10:10.677 align:middle position:46% size:68%
that brings the abcd matrix
into a diagonal form
00:10:10.677 --> 00:10:13.613 align:middle position:50% size:56%
by right-multiplying
the eigenvector matrix
00:10:13.613 --> 00:10:16.683 align:middle position:50% size:50%
and left-multiplying
its inverse.
00:10:16.683 --> 00:10:18.985 align:middle position:47% size:58%
This process is
called diagonalization,
00:10:18.985 --> 00:10:22.922 align:middle position:50% size:75%
which is handled by a computer
for a larger matrix.
00:10:22.922 --> 00:10:25.825 align:middle position:50% size:70%
Since the 70s chemists often
had the highest ability
00:10:25.825 --> 00:10:29.295 align:middle position:50% size:61%
to partially diagonalize
large matrices,
00:10:29.295 --> 00:10:31.631 align:middle position:48% size:48%
whose dimension can
exceed a billion,
00:10:31.631 --> 00:10:34.567 align:middle position:47% size:63%
in the context of quantum
chemical calculations.
00:10:35.869 --> 00:10:38.104 align:middle position:47% size:58%
This is the same matrix
eigenvalue problem
00:10:38.104 --> 00:10:39.305 align:middle position:50% size:56%
in the previous pages.
00:10:41.574 --> 00:10:44.878 align:middle position:50% size:81%
Subtracting the right-hand side
from both sides of the equation,
00:10:44.878 --> 00:10:46.646 align:middle position:50% size:61%
we obtain this equation.
00:10:48.615 --> 00:10:54.521 align:middle position:47% size:63%
Remember abcd are known,
but e and xy are unknown.
00:10:57.190 --> 00:11:00.493 align:middle position:50% size:75%
Now, there are two possible
cases concerning the existence
00:11:00.493 --> 00:11:02.428 align:middle position:50% size:75%
of the inverse of this matrix.
00:11:03.963 --> 00:11:06.599 align:middle position:44% size:78%
In case #1, the inverse exists.
00:11:06.599 --> 00:11:10.003 align:middle position:50% size:81%
Then we can multiply the inverse
from the left on both sides
00:11:10.003 --> 00:11:13.039 align:middle position:44% size:78%
of this equation
and we inevitably always obtain
00:11:13.039 --> 00:11:17.043 align:middle position:46% size:68%
this trivial zero vector
solution as an eigenvector.
00:11:17.043 --> 00:11:20.346 align:middle position:44% size:78%
Since the n-by-n matrix
eigenvalue equation should have
00:11:20.346 --> 00:11:24.784 align:middle position:45% size:73%
n nontrivial solutions,
this case should never happen
00:11:24.784 --> 00:11:26.753 align:middle position:48% size:48%
in the first place.
00:11:26.753 --> 00:11:30.290 align:middle position:47% size:58%
This means that case #2
should always happen,
00:11:30.290 --> 00:11:33.526 align:middle position:50% size:75%
in which the inverse
of this matrix does not exist,
00:11:33.526 --> 00:11:34.661 align:middle position:48% size:48%
which in turn means
00:11:34.661 --> 00:11:38.097 align:middle position:50% size:50%
that the determinant
of the matrix is 0.
00:11:38.097 --> 00:11:40.133 align:middle position:50% size:75%
In this 2-by-2 matrix example,
00:11:40.133 --> 00:11:42.435 align:middle position:48% size:43%
the determinant
is given by this.
00:11:44.137 --> 00:11:49.475 align:middle position:50% size:56%
Again, abcd are known,
but e is unknown.
00:11:49.475 --> 00:11:50.643 align:middle position:47% size:53%
So we have found that
00:11:50.643 --> 00:11:54.113 align:middle position:50% size:70%
the 2-by-2 matrix eigenvalue
equation is equivalent
00:11:54.113 --> 00:11:57.417 align:middle position:47% size:58%
to a quadratic equation
for eigenvalue e.
00:11:57.417 --> 00:12:00.920 align:middle position:45% size:73%
In general,
n-by-n matrix diagonalization
00:12:00.920 --> 00:12:04.724 align:middle position:50% size:75%
is equivalent to solving
nth-order polynomial equation,
00:12:04.724 --> 00:12:08.094 align:middle position:50% size:45%
both of which are
equally difficult.
00:12:08.094 --> 00:12:10.263 align:middle position:50% size:56%
Quantitative theories
of many science
00:12:10.263 --> 00:12:11.764 align:middle position:46% size:68%
and engineering disciplines
00:12:11.764 --> 00:12:14.701 align:middle position:50% size:65%
are expressed
by differential equations.
00:12:14.701 --> 00:12:16.603 align:middle position:47% size:53%
If these differential
equations are
00:12:16.603 --> 00:12:18.304 align:middle position:50% size:45%
of higher-than-one
dimensional,
00:12:18.304 --> 00:12:20.673 align:middle position:50% size:50%
derivative operators
in these equations
00:12:20.673 --> 00:12:23.109 align:middle position:50% size:75%
are usually partial derivative
operators
00:12:23.109 --> 00:12:26.346 align:middle position:47% size:58%
and not full derivative
operators.
00:12:26.346 --> 00:12:29.015 align:middle position:46% size:68%
In chemistry, the dimension
is almost always
00:12:29.015 --> 00:12:32.085 align:middle position:47% size:58%
much higher than one,
we might as well switch
00:12:32.085 --> 00:12:34.454 align:middle position:47% size:58%
our default derivatives
from full derivatives
00:12:34.454 --> 00:12:37.557 align:middle position:50% size:56%
to partial derivatives
from now on.
00:12:37.557 --> 00:12:39.525 align:middle position:44% size:78%
A partial derivative means that
we consider
00:12:39.525 --> 00:12:41.294 align:middle position:50% size:70%
only the explicit dependence
00:12:41.294 --> 00:12:45.565 align:middle position:50% size:81%
of the function f with respect
to the differentiating variable,
00:12:45.565 --> 00:12:48.301 align:middle position:50% size:41%
in this case, z.
00:12:48.301 --> 00:12:51.571 align:middle position:45% size:73%
There may be other variables,
x and y, which may somehow
00:12:51.571 --> 00:12:53.139 align:middle position:48% size:43%
be coupled with z
00:12:53.139 --> 00:12:58.111 align:middle position:45% size:73%
and can change the value of f
through changes in x and y.
00:12:58.111 --> 00:13:01.614 align:middle position:47% size:63%
These implicit dependence
is purposefully neglected
00:13:01.614 --> 00:13:03.683 align:middle position:50% size:65%
in the partial derivative.
00:13:03.683 --> 00:13:06.386 align:middle position:44% size:78%
In other words, when evaluating
the partial derivative
00:13:06.386 --> 00:13:09.322 align:middle position:47% size:58%
with respect to z,
we hold other variables
00:13:09.322 --> 00:13:11.858 align:middle position:50% size:56%
such as x and y fixed,
00:13:11.858 --> 00:13:15.728 align:middle position:45% size:73%
and these fixed variables are
indicated as subscripts here.
00:13:18.064 --> 00:13:20.967 align:middle position:47% size:53%
Consider a function f
of space variable, x,
00:13:20.967 --> 00:13:23.269 align:middle position:47% size:53%
and time variable, t.
00:13:23.269 --> 00:13:26.305 align:middle position:50% size:61%
Let the space variable x
also depend on time,
00:13:26.305 --> 00:13:29.475 align:middle position:50% size:70%
and so x is a function of t.
00:13:29.475 --> 00:13:32.045 align:middle position:50% size:61%
The full derivative of f
with respect to t
00:13:32.045 --> 00:13:36.816 align:middle position:50% size:75%
must take into account all
possible dependence of f on t.
00:13:36.816 --> 00:13:39.719 align:middle position:50% size:56%
That is the explicit
and direct change in f
00:13:39.719 --> 00:13:42.922 align:middle position:50% size:56%
upon change in t
while holding x fixed,
00:13:42.922 --> 00:13:45.858 align:middle position:44% size:78%
plus, the change mediated by x,
00:13:45.858 --> 00:13:49.962 align:middle position:50% size:81%
which is the product of the rate
of change with f with x
00:13:49.962 --> 00:13:53.599 align:middle position:50% size:56%
and the rate of change
of x with t.
00:13:53.599 --> 00:13:55.735 align:middle position:45% size:73%
Clearly, this full derivative
is different
00:13:55.735 --> 00:13:59.072 align:middle position:46% size:68%
from the partial derivative
of f with t.
00:13:59.072 --> 00:14:03.076 align:middle position:50% size:81%
In fact, partial derivatives are
simpler than full derivatives,
00:14:03.076 --> 00:14:05.645 align:middle position:48% size:48%
and we should feel
fortunate for that.
00:14:05.645 --> 00:14:08.314 align:middle position:50% size:65%
Other than this difference
in definition,
00:14:08.314 --> 00:14:12.652 align:middle position:44% size:78%
usual rules of differentiations
apply to partial derivatives.
00:14:12.652 --> 00:14:15.988 align:middle position:45% size:73%
For example, the partial
derivative of a sine function
00:14:15.988 --> 00:14:17.757 align:middle position:47% size:53%
is a cosine function,
00:14:17.757 --> 00:14:20.126 align:middle position:50% size:56%
the partial derivative
of an exponential is
00:14:20.126 --> 00:14:22.762 align:middle position:50% size:65%
an exponential, and so on.
00:14:22.762 --> 00:14:25.231 align:middle position:50% size:45%
However, there are
some pitfalls.
00:14:25.231 --> 00:14:30.169 align:middle position:45% size:73%
We know that this identity
is true for full derivatives.
00:14:30.169 --> 00:14:31.637 align:middle position:45% size:73%
Is the corresponding identify
00:14:31.637 --> 00:14:33.473 align:middle position:50% size:61%
also true
for partial derivatives?
00:14:35.041 --> 00:14:38.478 align:middle position:50% size:65%
The answer is yes and no,
depending on the variables
00:14:38.478 --> 00:14:40.379 align:middle position:48% size:43%
being held fixed.
00:14:40.379 --> 00:14:43.649 align:middle position:50% size:70%
It is a yes, if the variable
held fixed are the same
00:14:43.649 --> 00:14:46.185 align:middle position:47% size:53%
between the left-
and right-hand sides,
00:14:46.185 --> 00:14:49.021 align:middle position:50% size:56%
but it is a no,
if they are different.
00:14:49.021 --> 00:14:52.091 align:middle position:50% size:70%
So this is a trick question,
but a very common question
00:14:52.091 --> 00:14:54.260 align:middle position:45% size:73%
that you might subconsciously
ask yourself
00:14:54.260 --> 00:14:56.262 align:middle position:46% size:68%
and incorrectly answer yes.
00:14:57.029 --> 00:15:00.233 align:middle position:50% size:75%
So until you become an expert,
you might want to make a habit
00:15:00.233 --> 00:15:02.935 align:middle position:50% size:75%
of writing the fixed variables
explicitly
00:15:02.935 --> 00:15:05.071 align:middle position:45% size:73%
to guard against such errors.
00:15:05.071 --> 00:15:08.107 align:middle position:50% size:75%
Let us consider higher partial
derivatives.
00:15:08.508 --> 00:15:12.645 align:middle position:46% size:68%
Suppose we have
a function, f(x, y, and z,)
00:15:12.645 --> 00:15:16.115 align:middle position:50% size:65%
which is to be partial-
differentiated by x and y.
00:15:16.115 --> 00:15:18.084 align:middle position:50% size:65%
Does the order
of differentiation matter?
00:15:19.085 --> 00:15:20.753 align:middle position:48% size:43%
The answer is no.
00:15:20.753 --> 00:15:24.023 align:middle position:46% size:68%
It does not matter if you
differentiate with y first,
00:15:24.023 --> 00:15:26.259 align:middle position:50% size:56%
holding x and z fixed,
00:15:26.259 --> 00:15:30.763 align:middle position:44% size:78%
then differentiate with x next,
holding y and z fixed,
00:15:30.763 --> 00:15:36.636 align:middle position:50% size:75%
or differentiate with x first,
holding y and z fixed,
00:15:36.636 --> 00:15:40.606 align:middle position:44% size:78%
then differentiate with y next,
holding x and z fixed.
00:15:41.874 --> 00:15:44.644 align:middle position:50% size:65%
These two orders
of partial differentiation
00:15:44.644 --> 00:15:49.115 align:middle position:45% size:73%
merely differ in the path
they take in varying x and y,
00:15:49.115 --> 00:15:50.750 align:middle position:50% size:75%
but have the same destination.
00:15:53.052 --> 00:15:54.453 align:middle position:50% size:65%
And so a higher derivative
00:15:54.453 --> 00:15:58.624 align:middle position:46% size:68%
is invariant with the order
of differentiations.
00:15:58.624 --> 00:16:00.660 align:middle position:44% size:78%
Again we should feel fortunate.
00:16:01.227 --> 00:16:03.963 align:middle position:47% size:63%
Let us revisit
the Schrodinger equation.
00:16:03.963 --> 00:16:06.098 align:middle position:47% size:53%
The many derivative
operators that appear
00:16:06.098 --> 00:16:07.767 align:middle position:46% size:68%
in the Schrodinger equation
00:16:07.767 --> 00:16:11.337 align:middle position:45% size:73%
should be partial derivatives
and not full derivatives.
00:16:11.337 --> 00:16:14.774 align:middle position:50% size:75%
So, the time-dependent
Schrodinger equation should be
00:16:14.774 --> 00:16:16.943 align:middle position:50% size:56%
more precisely written
like this.
00:16:19.679 --> 00:16:21.881 align:middle position:50% size:56%
The time derivative is
a partial derivative
00:16:21.881 --> 00:16:27.019 align:middle position:50% size:75%
which holds all spatial
variables, x, y, and z, fixed.
00:16:27.019 --> 00:16:29.889 align:middle position:50% size:65%
From now on, we may always
use partial derivatives
00:16:29.889 --> 00:16:31.123 align:middle position:47% size:53%
with respect to time.
00:16:32.358 --> 00:16:35.761 align:middle position:50% size:70%
Time-independent Schrodinger
equation in 1-dimension
00:16:35.761 --> 00:16:36.929 align:middle position:47% size:53%
is written like this.
00:16:38.030 --> 00:16:40.766 align:middle position:50% size:70%
I trust that you can
write this down from memory.
00:16:40.766 --> 00:16:43.769 align:middle position:50% size:56%
Since this is truly
1-dimensional equation
00:16:43.769 --> 00:16:46.839 align:middle position:44% size:78%
and there is no other variables
to be held fixed,
00:16:46.839 --> 00:16:50.042 align:middle position:45% size:73%
it does not matter if we used
partial or full derivatives
00:16:50.042 --> 00:16:52.945 align:middle position:47% size:53%
in this exceptionally
tiny problem.
00:16:52.945 --> 00:16:56.215 align:middle position:50% size:65%
However, it does not hurt
to use partial derivatives
00:16:56.215 --> 00:16:57.850 align:middle position:50% size:65%
for notational uniformity.
00:16:59.285 --> 00:17:02.154 align:middle position:46% size:68%
If the particle can move
in the 3-dimensional space,
00:17:02.154 --> 00:17:06.292 align:middle position:50% size:81%
there are 3 spatial coordinates,
x, y, and z,
00:17:06.292 --> 00:17:08.194 align:middle position:50% size:70%
in the Schrodinger equation.
00:17:08.194 --> 00:17:10.263 align:middle position:47% size:63%
Then, the three Cartesian
coordinates
00:17:10.263 --> 00:17:12.465 align:middle position:50% size:56%
of the linear momentum
operator
00:17:12.465 --> 00:17:14.834 align:middle position:50% size:61%
must be defined
with partial derivatives
00:17:14.834 --> 00:17:16.502 align:middle position:50% size:75%
and not with full derivatives.
00:17:18.204 --> 00:17:22.074 align:middle position:44% size:78%
For example, x-component linear
momentum operator is
00:17:22.074 --> 00:17:29.682 align:middle position:45% size:73%
–iħ(∂/∂x) which holds y and z
fixed during differentiation.
00:17:32.485 --> 00:17:35.788 align:middle position:50% size:75%
We can define the linear
momentum vector operator using
00:17:35.788 --> 00:17:39.592 align:middle position:47% size:58%
this vector partial
derivative operator, ∇.
00:17:39.592 --> 00:17:45.598 align:middle position:50% size:70%
This –iħ∇ is an operator
that consists of 3 elements,
00:17:45.598 --> 00:17:50.303 align:middle position:44% size:78%
each of which is an operator
involving a partial derivative.
00:17:50.303 --> 00:17:52.438 align:middle position:47% size:53%
When it acts
on an input function,
00:17:52.438 --> 00:17:54.907 align:middle position:50% size:56%
it returns a set
of 3 output functions.
00:17:56.609 --> 00:17:58.311 align:middle position:44% size:78%
Classically, the kinetic energy
00:17:58.311 --> 00:18:03.015 align:middle position:45% size:73%
is the sum of the x, y, and z
component kinetic energies.
00:18:03.015 --> 00:18:05.718 align:middle position:46% size:68%
In quantum mechanics,
the kinetic energy operator
00:18:05.718 --> 00:18:08.487 align:middle position:45% size:73%
is the sum of the x, y, and z
00:18:08.487 --> 00:18:11.257 align:middle position:48% size:43%
component kinetic
energy operators.
00:18:11.257 --> 00:18:13.259 align:middle position:45% size:73%
The x component, for example,
00:18:13.259 --> 00:18:17.029 align:middle position:45% size:73%
involves the partial second
derivative with respect to x,
00:18:17.029 --> 00:18:19.732 align:middle position:50% size:65%
which holds y and z fixed.
00:18:19.732 --> 00:18:22.668 align:middle position:50% size:81%
You might notice that
the rule of vector inner product
00:18:22.668 --> 00:18:25.838 align:middle position:50% size:65%
can be applied
to the vector operator, ∇,
00:18:25.838 --> 00:18:29.342 align:middle position:44% size:78%
to give this correct definition
of the kinetic-energy operator.
00:18:31.711 --> 00:18:35.381 align:middle position:50% size:75%
In other words, ∇ is
a 3-component vector operator,
00:18:35.381 --> 00:18:39.151 align:middle position:50% size:61%
whereas ∇^2 is the usual
1-component operator.
00:18:40.519 --> 00:18:43.923 align:middle position:46% size:68%
Some textbooks introduce
this new symbol, Laplacian,
00:18:43.923 --> 00:18:46.459 align:middle position:48% size:43%
to designate ∇^2.
00:18:46.459 --> 00:18:49.362 align:middle position:50% size:75%
We will not use it here,
but you might want to be aware
00:18:49.362 --> 00:18:50.529 align:middle position:48% size:43%
of this notation.
00:18:51.731 --> 00:18:54.033 align:middle position:47% size:58%
So the time-independent
Schrodinger equation
00:18:54.033 --> 00:18:57.970 align:middle position:50% size:81%
of one particle in 3 dimensional
Cartesian coodinates
00:18:57.970 --> 00:19:03.409 align:middle position:44% size:78%
is written more precisely like
so...using partial derivatives.
00:19:04.777 --> 00:19:08.114 align:middle position:45% size:73%
In many chemical applications
of the Schrodinger equation,
00:19:08.114 --> 00:19:11.217 align:middle position:50% size:81%
it is far more convenient to use
the spherical coordinates rather
00:19:11.217 --> 00:19:12.485 align:middle position:44% size:78%
than the Cartesian coordinates.
00:19:14.320 --> 00:19:17.857 align:middle position:44% size:78%
In the spherical coordinates,
we specify a point in the space
00:19:17.857 --> 00:19:20.226 align:middle position:50% size:81%
by the radius r from the origin,
00:19:20.226 --> 00:19:23.162 align:middle position:48% size:43%
the polar angle θ
from the z-axis,
00:19:23.162 --> 00:19:27.033 align:middle position:47% size:63%
and the azimuthal angle φ
from the x-axis.
00:19:27.033 --> 00:19:29.402 align:middle position:46% size:68%
You do not need to memorize
the exact definition
00:19:29.402 --> 00:19:31.337 align:middle position:44% size:78%
of these spherical coordinates,
00:19:31.337 --> 00:19:33.305 align:middle position:50% size:50%
but you might want
to be able to derive
00:19:33.305 --> 00:19:36.342 align:middle position:50% size:75%
these conversion formulas
from the spherical coordinates
00:19:36.342 --> 00:19:39.712 align:middle position:50% size:70%
to Cartesian coordinates,
when the figure is provided.
00:19:39.712 --> 00:19:42.548 align:middle position:50% size:70%
The Schrodinger equation
in the spherical coordinates
00:19:42.548 --> 00:19:44.517 align:middle position:47% size:53%
becomes significantly
more complicated.
00:19:46.252 --> 00:19:49.488 align:middle position:50% size:75%
This is the ∇^2
in the kinetic energy operator
00:19:49.488 --> 00:19:51.857 align:middle position:45% size:73%
in the Cartesian coordinates,
as we already know.
00:19:53.859 --> 00:19:57.963 align:middle position:45% size:73%
This is how the ∇^2 looks
in the spherical coordinates.
00:19:57.963 --> 00:20:01.534 align:middle position:50% size:75%
You most certainly do not have
to memorize this expression,
00:20:01.534 --> 00:20:03.569 align:middle position:50% size:70%
because you know where
to find it when you need it.
00:20:05.271 --> 00:20:07.540 align:middle position:46% size:68%
It is sufficient if you can
recognize it as part
00:20:07.540 --> 00:20:10.776 align:middle position:45% size:73%
of the Hamiltonian
in the spherical coordinates.
00:20:10.776 --> 00:20:14.246 align:middle position:45% size:73%
I assure you that the benefit
of subsequent simplifications
00:20:14.246 --> 00:20:17.750 align:middle position:50% size:75%
by the spherical coordinates
for some problems far outweigh
00:20:17.750 --> 00:20:21.053 align:middle position:47% size:63%
the initial complications
in the ∇^2 expression.
00:20:22.188 --> 00:20:24.790 align:middle position:48% size:48%
This is the second
challenge homework.
00:20:24.790 --> 00:20:28.194 align:middle position:44% size:78%
Derive the spherical-coordinate
expression of the ∇^2,
00:20:28.194 --> 00:20:30.262 align:middle position:50% size:65%
given in this green panel,
00:20:30.262 --> 00:20:32.431 align:middle position:44% size:78%
starting with the
Cartesian-coordinate expression
00:20:32.431 --> 00:20:34.100 align:middle position:48% size:43%
in the blue panel
00:20:34.100 --> 00:20:36.969 align:middle position:45% size:73%
and the coordinate conversion
in the orange panel.
00:20:38.737 --> 00:20:39.738 align:middle position:47% size:53%
The first thing to do
00:20:39.738 --> 00:20:42.141 align:middle position:46% size:68%
is to invert the coordinate
conversion formulas
00:20:42.141 --> 00:20:45.945 align:middle position:50% size:50%
in the orange panel,
so that r, θ, and φ
00:20:45.945 --> 00:20:49.448 align:middle position:47% size:58%
are expressed
in terms of x, y and z.
00:20:49.448 --> 00:20:52.218 align:middle position:50% size:61%
Without this step,
you may be stuck forever
00:20:52.218 --> 00:20:54.320 align:middle position:47% size:53%
and this is related
to the trick question
00:20:54.320 --> 00:20:57.556 align:middle position:50% size:61%
about the reciprocal
of a partial derivative.
00:20:57.556 --> 00:21:00.126 align:middle position:50% size:75%
Let me summarize this lecture.
00:21:00.126 --> 00:21:02.595 align:middle position:50% size:70%
We have reviewed some basic,
but essential concepts
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of mathematics
for quantum mechanics.
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It may be counterproductive
to proceed in this course
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before familiarizing yourself
with these concepts first.
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We have seen that
the derivative operators
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in the Schrodinger equations
are in fact partial derivatives,
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which are in many ways
simpler than full derivatives.
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We have seen the forms
of the kinetic-energy operators
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in the Cartesian coordinates and
in the spherical coordinates.