Schrodinger Equation (seq 2 Common Sense Quantum Physics)

Schrodinger Equation made easy.

Here's the videoscript.

Hello, I'm Arjen, the Common Sense Quantum Physicist. My goal is to develop intuitive approaches to Quantum Physics. In a first sequence, about the polarization of light, we saw that quantum particles and particularly those composing light, the photons, are best represented by spinning linear objects, like arrows or needles or rods.

This time, we'll look at how we may characterize the physics of arrows. A famous physicist, Paul Dirac, in this reference book on the Principles of Quantum Mechanics, proposed to call them ket vectors, or simply kets [k e t], and denote a general one of them by a special symbol | "GT". If we want to specify a particular one of them be a label, A say, we insert it in the middle, thus vertical bar A greater than sign.

So, a ket is nothing more than a rotating arrow... Well, nearly nothing more. Physicists perform operating rules on it. These rules tell us how an arrow is transformed into another arrow, these rules also tell us that you may add the arrows in order to get another arrow, or multiply them or subtract them one from another. Let me present some of these operations.

Firstly, there is the operation where the arrow is rotated by an angle alpha. We multiply ket |A"GT" by a so called complex number to describe this rotation: exp(i alpha). So when you see a complex number in quantum-mechanical expressions, it is real physics. It simply means that ket |A"GT" has undergone a rotation in its spinning surface.

Secondly, you may add arrows by positionning them head to tail and then imagining the resultant arrow.
So this arrow |A"GT" plus this arrow |B"GT" equals this arrow |C"GT".

Subtracting is easy, you just imagine the arrow pointing towards the other side. So, subtracting ket |B"GT" from ket |A"GT" is the same as adding arrow -|B"GT" to arrow |A"GT" and this gives arrow |D"GT".

Multiplying by a real number is also easy.
2 * arrow |A"GT" just means |A"GT"+|A"GT" and 0.5 * arrow |A"GT" is just the operation that halves the arrow |A"GT".

So now that we know some basics of computing with arrows, let us describe an ordinary situation, for example the rotating motion of this arrow.

We already saw that |B"GT" = exp(i alpha) * |A"GT" just means that in order to obtain arrow |B"GT", arrow |A"GT" has been rotated by an angle alpha.

If we know the constant angular speed at which this arrow spins, we could replace the angle alpha by the expression (angular speed omega * interval of time). So we could also say:

Arrow |B"GT" equates exp(i * angular speed omega * difference (final time -- initial time)) * arrow |A"GT"
[written: |B"GT" = exp(i omega Delta(t)) |A"GT"]
This is the time evolution law for a freely spinning arrow.

However this is valid only for arrows that spin with a constant angular speed. So this equation has a very limited domain of validity. In real life a rotating linear object is generally subject to various forces. This arrow is subject to the forces of my hand, the motion of the blades of a windmill depends on the force of the wind and the motion of the spokes of this bicycle is bound to the motion impinged on the wheel.

So how could we describe the physics in the general case? Well, the trick is to look at very tiny intervals of time, which is called a differential of time and written dt. After a very tiny interval of time, the arrow |A"GT" is rotated over a very tiny angle omega * dt. The difference between the two very close positions of arrow |A"GT" is also a differential. Remember the head to tail rule. It is the little arrow d|A"GT" that is obtained by subtracting the initial |A"GT" from the subsequent |A"GT".

But there is another way to write the little arrow d|A"GT". Imagine the time interval going to the limit 0. Then the tiny arrow d|A"GT" makes an angle of 90° with |A"GT". This means that you may obtain the arrow d|A"GT" from |A"GT" not only by subtraction but also through a rotation of 90° and a reduction of the arrow by the proportionality factor (instantaneous angular speed omega * dt). That's just multiplying arrow |A"GT" by exp (i 90°) * instantaneous angular speed * dt. Well, you may have learned at high school that exp (i 90°) is the same as the imaginary i. So we may write in compact form:

d|A"GT"= i omega * dt |A"GT"

which is the generalized form of Schrodinger's time-dependent equation. Physicists generally multiply both terms of this equation by a fundamental constant hbar and reverse the time direction. This equation just describes common sense physics of ordinary arrow-like objects, for which the rotational speed at the tips of the arrow always make an angle of 90° with the arrow.

Now, when you ride your bike, remember the spokes obey the Schrodinger equation.

Next time we'll look at how arrow-like objects interact.

Category:

Science & Technology

Tags:

• Quantum
• Mechanics
• Physics
• Schrodinger
• Equation
• explained
• Hamiltonian
• Dirac
• Planck
• Ket
• Vector
• Arrow
• Arjen
• Dijksman

[MegaPhysicist]

hbar x omega is a number H is an operator. And your arrow notation is nothing new, the kets belong to a Hilbert Space.

[ArjenDijksman]

True. An operator operating on a ket just gives a number times that ket, a number given the measure of the motion/energy of the object represented by the ket.

[EMOjamesy23]

@ArjenDijksman

more specifically a linear operator acting on a ket vector gives a number times that ket, operators are not alway linear

[ArjenDijksman]

@EMOjamesy23 That's true, the linear operator is a specific case of the quantum operator.

[henrikhhhhh]

this is pretty classical stuff, but with ket notation, so its not really quantum mechanical.

Rotating vectors is not the difficult part of QM.

[ArjenDijksman]

Classically a rotating arrow would be described by moment of inertia (integral over all point masses * square distance from axis) + angular velocity, never with a ket vector. Quantum mechanics don't go in the details of definite locations but considers the arrow as a whole and represents it as a vector. You get simpler basics with Quantum Mechanics but complex computations due to the probability distributions of all those spinning arrows interacting with each other.

[eggertstwart]

@FelipeZucchetti check-out should not be hyphenated.

[FelipeZucchetti]

Hello mate, if you want, check-out my channel...:)

[nickharvey7]

Interesting video!

In my video The Paradox of Schrodingers Cat an artist view Time has symmetry and geometry could this explaining the paradoxes of Quantum Physics? Could time form continuously by the emission and absorption of light or EMR from one atom to another continuously forming the broken symmetry of spacetime.

[ArjenDijksman]

You're right. In fact, I inserted it to show that you could insert any constant at both sides and the equation is still valid. hbar is a constant that translates angular frequencies into energies, in our unit system.

[xniquilador]

Thanks Arjen; I think Dirac was of the most exceptional visionaries in 'the field'.

[citrusRF]

I love physics when I was in HS