วิถีการเบี่ยงของ electron ในการทดลองของ Rutherford และ ฟังก์ชันคลื่นกับความน่าจะเป็นที่จะพบอิเล็กตรอน ณ จุดๆหนึ่ง บนกราฟ 3 มิติ

particle diagram ******

จำได้ไหมเอ่ย ตอนสอบ Summative (สอบกลางภาค) ครั้งแรกของม.4 วิชาเคมี อ.ได้ให้อธิบายการทดลองของ Rutherford ซึ่งเฉลยว่า ใครเขียนคำว่า ชน ไม่ให้คะแนน ซึ่งจากภาพ เฉลยก็น่าจะถูกน่ะ จากจลน์ศาสตร์เคมี ที่มีทฤษฎีการชนที่ถูกทิศทางทำให้เกิด Reaction โดยการชนให้เกิดพันธะเคมีคือ มี 90 % ของ Orbital มาเริ่ม Overlap กัน แต่อย่าลืมว่า นี้คือการชนระหว่าง Particle ไม่ใช่ระหว่าง Atom 55+

https://i1.wp.com/hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/hoscom2.gif

กราฟของ Wave Function ที่ก่อนผ่านการยกกำลังสอง ทำให้ผลจากค่า Function ไม่ติดลบ
ต่อไปก็ลองอ่านดูน่ะ เราว่า เราต้องเรียน Complex Number กันก่อน แล้วเราเรียนแคลคูลัสเกี่ยว Complex Function น่ะถึงจะรู้เรื่อง ตอนนี้เราก็ไม่รู้เรื่อง 55+

Complex Conjugate

The conjugate of a complex number is that number with the sign of the imaginary part reversed

The utility of the conjugate is that any complex number multiplied by its complex conjugate is a real number:

This operation has practical utility for the rationalization
of complex numbers and the square root of the number times its
conjugate is the magnitude of the complex number when expressed in polar form.

Applications of Complex Conjugates

When a real positive definite quantity is needed from a real
function, the square of the function can be used. In the case of a
complex function, the complex conjugate
is used to accomplish that purpose. The product of a complex number and
its complex conjugate is the complex number analog to squaring a real
function. The complex conjugate is used in the rationalization of complex numbers and for finding the amplitude of the polar form of a complex number.

One application of the complex conjugate in physics is in finding the probability in quantum mechanics. Since the wavefunction
which defines the probability amplitude may be a complex function, the
probability is defined in terms of the complex conjugate to obtain a
real value.

Probability in Quantum Mechanics

The wavefunction
represents the probability amplitude for finding a particle at a given
point in space at a given time. The actual probability of finding the
particle is given by the product of the wavefunction with it’s complex conjugate (like the square of the amplitude for a complex function).



Since the probability must be = 1 for finding the particle somewhere,
the wavefunction must be normalized. That is, the sum of the
probabilities for all of space must be equal to one. This is expressed
by the integral

Examples of normalization

Part of a working solution to the Schrodinger equation is the
normalization of the solution to obtain the physically applicable
probability amplitudes.

ธรรมชาติของฟังก์ชันเชิงซ้อน

In quantum mechanics, a probability amplitude is a complex number whose modulus squared represents a probability or probability density. For example, the values taken by a normalised wave function ψ are amplitudes, since |ψ(x)|2 gives the probability density at position x. Probability amplitudes may also correspond to probabilities of discrete outcomes.

The principal use of probability amplitudes is as the physical meaning of the wavefunction, a link first proposed by Max Born and a pillar of the Copenhagen interpretation
of quantum mechanics. In fact, the properties of the wave function were
being used to make physical predictions (such as emissions from atoms
being at certain discrete energies) before any physical interpretation
was offered. Born was awarded half of the 1954 Nobel Prize in physics
for this understanding, though it was vigorously contested at the time
by the original physicists working on the theory, such as Schrödinger and Einstein.
Therefore, the probability thus calculated is sometimes called the
“Born probability”, and the relationship used to calculate probability
from the wavefunction is sometimes called the Born rule.

These probability amplitudes have special significance because they
act in quantum mechanics as the equivalent of conventional
probabilities, with many analogous laws. For example, in the classic double-slit experiment where electrons are fired randomly at two slits, an intuitive interpretation is that P(hit either slit) = P(hit first slit) + P(hit second slit), where P(event)
is the probability of that event. However, it is impossible to observe
which slit is passed through without altering the electron. Thus, when
not watching the electron, the particle cannot be said to go through
either slit and this simplistic explanation does not work. However, the
complex amplitudes taken by the two wavefunctions which represent the
electron passing each slit do follow a law of precisely the form
expected (ψtotal =ψfirst + ψsecond), and the calculations agree with experiment. This is the principle of quantum superposition,
and explains the requirement that amplitudes be complex, as a purely
real formulation has too few dimensions to describe the system’s state
when superposition is taken into account.[1]

บางครั้ง ถ้าเราเข้าใจกราฟจำนวนเชิงซ้อน อาจนำไปสู่การแก้ปัญหาคาใจได้ครับ

Credit: http://hyperphysics.phy-astr.gsu.edu/hbase/cmplx2.html#c3
Help Credit: http://en.wikipedia.org/wiki/Complex_function and http://e-book.ram.edu/e-book/m/MA217/MA217-2.pdf and http://www.math.ksu.edu/~bennett/jomacg/ For Learn Complex Function

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s