Physics - Ch 66 Ch 4 Quantum Mechanics: Schrodinger Eqn (6 of 92) The Schrodinger Eqn. in 1-D (2/3)

Michel van Biezen
29 Jan 201705:28
EducationalLearning
32 Likes 10 Comments

TLDRThe video script delves into the quantum mechanics concepts of energy and momentum, particularly for photons and particles with mass. It explains the derivation of equations for the energy and momentum of a photon, and then transitions to discussing the energy of a massive particle, combining kinetic and potential energy. The script highlights the leap to equating the energy of a particle with the energy of a photon, leading to the development of the Schrödinger equation. The process of deducing this fundamental equation is described as a blend of educated guessing and the need for it to align with known wave-like behaviors of particles. The video sets the stage for further exploration in the subsequent episode, where the Schrödinger equation's solutions and their implications will be discussed.

Takeaways
  • 🌟 The energy of a photon is described by the equation E = H bar * Omega, where Omega is the angular frequency.
  • 🚀 The momentum of a photon is given by the equation P = H bar * K, with H bar being h / 2 pi and h being Planck's constant.
  • 🔍 The energy of a massive particle is the sum of its kinetic and potential energy, with kinetic energy expressed as (1/2) * m * v^2 and potential energy represented by V.
  • 🌊 Small particles exhibit wave-like properties similar to photons, leading to the substitution of energy and momentum in their respective equations.
  • 📝 The guessed equation for the energy of a particle with mass is E = H bar * Omega = - (H bar^2 * k^2) / (2 * m) + V, which is only valid for particles with mass.
  • 🤔 The process of deriving the energy equation involved educated guessing and multiple attempts, eventually leading to a correct description of a particle's wave.
  • 🧠 The Schrödinger equation was initially guessed to have a similar form to the wave equation, but with modifications to account for the particle's potential energy.
  • 📐 The guessed form of the Schrödinger equation is ∂ψ/∂t = (-H bar^2 / (2 * m)) * ∂^2ψ/∂x^2 + V * ψ, where ψ is the wave function.
  • 🌈 The traditional wave solutions like sin(kx - Omega t) or cos(kx - Omega t) do not work with the new equation due to the first partial derivative term.
  • 🔄 The approach to finding the differential equation was to work backwards from a guessed solution format to determine the equation's structure.
  • 🔗 The constants in the equations are derived from the relationship between photons and small particles, reflecting the quantum nature of reality.
Q & A
  • What are the three equations discussed in the video?

    -The three equations discussed are: 1) The energy of a photon, written as H bar times Omega (angular frequency), 2) The momentum of a photon, written as H bar times K, and 3) An equation describing the energy of a particle with mass, which is the sum of its kinetic and potential energy.

  • How is the energy of a photon related to its momentum and the speed of light?

    -The energy of a photon is the product of its momentum and the speed of light (E = p * c).

  • What is the kinetic energy of a particle with mass?

    -The kinetic energy of a particle with mass is given by the formula (1/2) * m * v^2, where m is the mass and v is the velocity of the particle.

  • How does the potential energy of a particle influence its total energy?

    -The total energy of a particle with mass is the sum of its kinetic energy and its potential energy, represented by the symbol V.

  • What is the significance of the substitution of the energy in the equation with H bar Omega?

    -The substitution of the energy in the equation with H bar Omega is a leap of faith based on the idea that small particles, like photons, exhibit wave-like properties, and this allows for the description of the wave-like behavior of particles such as electrons.

  • Why is the energy equation for a particle with mass only valid for real particles and not for photons?

    -The energy equation for a particle with mass is only valid for real particles because photons, being massless, do not have potential energy in the same sense as particles with mass.

  • What is the Schrödinger equation mentioned in the script?

    -The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes with time.

  • How did the scientists arrive at the form of the Schrödinger equation?

    -The scientists used educated guessing and worked backwards from the known solutions of the wave equation to formulate the Schrödinger equation.

  • What is the role of the constant H bar in the equations?

    -H bar, which is Planck's constant divided by 2 pi (h / 2 pi), is a fundamental constant that appears in the equations describing the energy and momentum of photons and the energy of particles with mass.

  • Why did the scientists replace the momentum P with H bar k in the energy equation?

    -They replaced the momentum P with H bar k to incorporate the relationship between the momentum of a particle and its wave-like properties, as derived from the de Broglie hypothesis.

  • What is the significance of the wave function in the context of the Schrödinger equation?

    -The wave function, which is a solution to the Schrödinger equation, provides information about the probability amplitude of finding a particle in a particular state or position.

  • How does the video script relate to the understanding of quantum mechanics?

    -The script provides a foundational understanding of the mathematical constructs and principles that underpin quantum mechanics, such as the relationship between energy, momentum, and wave-like behavior of particles.

Outlines
00:00
🌟 Quantum Mechanics Foundations: Energy and Momentum of Photons

This paragraph introduces the basic concepts of quantum mechanics, focusing on the energy and momentum of photons. It explains that the energy of a photon is given by the product of Planck's constant (H bar) and the angular frequency (Omega), and the momentum is given by H bar times the wave number (K). The discussion then transitions to finding an equation for the energy of a particle with mass, which includes both kinetic and potential energy. The kinetic energy is expressed as 1/2 m v^2, and the potential energy is represented by V. The paragraph also describes how the energy and momentum of a photon are related to the energy and momentum of a particle, leading to the development of a new equation for the energy of a particle with mass.

05:01
📚 Deriving the Schrödinger Equation: The Quantum Leap

This paragraph delves into the process of deriving the Schrödinger equation, which describes the wave function of a particle. It starts by discussing the educated guesses and the iterative process that led to the development of the equation. The paragraph explains how the energy and momentum of a particle are substituted into the equation, leading to the famous Schrödinger equation in the form of h bar^2 k^2 / 2m plus V equals the product of Planck's constant and the second derivative of the wave function with respect to position, plus the potential energy times the wave function. It also touches on the limitations of the equation, noting that it is only valid for particles with mass and not for massless particles like photons. The paragraph concludes by hinting at the next steps, which involve exploring the solutions to the Schrödinger equation and understanding why certain solutions work while others do not.

Mindmap
Linearity of the Wave Equation
Potential Energy
Quantum Mechanics
Wave-Particle Duality
Limitations
Wave Function
Schrödinger Equation
Gross and Zener's Contributions
Initial Assumptions
Substitution in Energy Equation
Momentum
Kinetic Energy
Total Energy
Relationship between Energy and Momentum
Momentum of a Photon
Energy of a Photon
Implications and Applications
Development of the Schrödinger Equation
Energy and Momentum of Massive Particles
Photon Properties
Quantum Mechanics and Wave-Particle Duality
Alert
Keywords
💡Photon
A photon is a particle representing a quantum of light, electromagnetic radiation, or electromagnetic energy. In the video, photons are discussed in the context of their energy and momentum, which are described by specific equations. The energy of a photon is given by the product of the reduced Planck constant (ħ) and the angular frequency (Ω), and the momentum is given by ħ times the wave vector (k).
💡Energy
In the context of the video, energy refers to the property of a system that can be transferred or converted into other forms, such as kinetic or potential energy. For photons, energy is quantized and described by the equation E = ħΩ. For particles with mass, the energy is the sum of kinetic and potential energy, with kinetic energy given by the formula 1/2mv^2 and potential energy represented by V.
💡Momentum
Momentum is a measure of the quantity of motion of an object and is a vector quantity that depends on both the mass and velocity of the object. In the video, the momentum of a photon is described by the equation p = ħk, where ħ is the reduced Planck constant and k is the wave vector. For particles with mass, momentum is related to velocity and mass, as shown by the equation p = mv.
💡Reduced Planck Constant (ħ)
The reduced Planck constant (ħ) is a fundamental constant in quantum mechanics, denoted as ħ = h / (2π), where h is Planck's constant. It is used to describe the quantization of energy and momentum in quantum systems. In the video, ħ is used in the equations for the energy and momentum of photons, as well as in the equations derived for particles with mass.
💡Angular Frequency (Ω)
Angular frequency (Ω) is a measure of how quickly an object rotates or oscillates around a central point, and is related to the frequency by the equation Ω = 2πf, where f is the frequency. In the video, angular frequency is used in the energy equation for a photon, E = ħΩ, to describe the energy associated with the photon's oscillation.
💡Potential Energy (V)
Potential energy is the stored energy of an object based on its position or configuration. In the video, potential energy is considered as part of the total energy of a particle with mass, in addition to its kinetic energy. It is represented by the letter V and is included in the energy equation for a particle with mass.
💡Kinetic Energy
Kinetic energy is the energy that a body possesses due to its motion. In the video, kinetic energy is discussed in the context of a particle with mass, using the formula KE = 1/2mv^2, where m is the mass and v is the velocity of the particle. It is one part of the total energy of the particle, along with potential energy.
💡Wave Equation
A wave equation is a mathematical equation that describes the behavior of waves, such as the propagation of light or sound. In the video, the wave equation is derived for a particle with mass, taking into account both its wave-like properties and its energy. The equation is linear and has a specific form that allows for solutions representing wave functions.
💡Schrödinger Equation
The Schrödinger Equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes with time. In the video, the Schrödinger Equation is introduced as the differential equation that governs the behavior of particles with mass, incorporating concepts from the energy and momentum of photons.
💡Quantum Mechanics
Quantum mechanics is a branch of physics that deals with the behavior of particles at the atomic and subatomic level. It is the foundation for understanding the wave-particle duality and the probabilistic nature of particles. The video discusses concepts and equations that are central to quantum mechanics, such as the energy and momentum of photons and the behavior of particles with mass.
💡Wave-Particle Duality
Wave-particle duality is a fundamental concept in quantum mechanics that states that every particle, such as an electron, has properties of both a wave and a particle. In the video, this concept is implied through the discussion of how small particles like electrons can be described by wave equations, similar to photons.
Highlights

The video discusses the derivation of equations for the energy and momentum of a photon.

The energy of a photon is given by the equation E = H bar * Omega, where Omega is the angular frequency.

The momentum of a photon is described by the equation P = H bar * K, with H bar being h / 2 pi and H being Planck's constant.

The video then explores the energy of a particle with mass, which is the sum of its kinetic and potential energy.

The kinetic energy is expressed as (1/2) * m * v^2, and the potential energy is represented by the letter V.

A leap is made by substituting the energy in the kinetic energy equation with H bar Omega, based on the wave-like behavior of small particles.

The resulting equation for the energy of a particle with mass is H bar Omega = (p^2 / 2m) + V, where p is the momentum.

This equation is only valid for particles with mass, as photons do not have potential energy.

The video introduces the Schrödinger equation as a differential equation to describe particles in one dimension.

The Schrödinger equation is hypothesized to be a linear equation, solvable as the sum of two linear equations.

The first guess for the Schrödinger equation is derived from educated guessing and the known relationship between photons and small particles.

The video explains that the traditional wave equation solutions, such as sin(kx - Omega t), no longer apply to the new equation.

A new solution format is proposed that works with the Schrödinger equation, which will be explored in the next video.

The process of deriving the Schrödinger equation is based on working backwards from a desired solution format.

The constants in the Schrödinger equation are derived from the relationship between photons and small particles.

The video sets the stage for the next installment, where the derivation and validation of the Schrödinger equation will be further explored.

Transcripts
00:00

welcome to elector on line in the

00:03

previous video we were able to come up

00:04

with three equations two of them that

00:07

describe the energy of a photon the

00:09

third one that describe the momentum of

00:12

a photon energy of a photon can be

00:14

written as H bar time Omega which is the

00:16

angular frequency the energy of a photon

00:19

can be written as the product of the

00:20

momentum and the speed of light and the

00:23

momentum of a photon can be written as H

00:26

bar * K remember H bar is h / 2 pi H

00:31

being Plank's

00:33

constant next we're trying to find an

00:35

equation that describes the energy of a

00:38

particle that has mass which means the

00:40

energy will be the sum of the kinetic

00:42

energy plus the potential energy of that

00:45

particle the kinetic energy will be 1/

00:48

12 m v^ s the potential energy can be

00:50

represented by the letter

00:52

V now what we can do is we can multiply

00:57

the numerator by m and divide the

00:59

numerator by m the reason why we do that

01:02

is then in the numerator we get M * V

01:05

squared and M * V is the momentum so

01:08

this can now be written as the momentum

01:11

squared / 2 * m this would be the

01:14

kinetic energy of the particle plus the

01:18

potential energy of the

01:21

particle now since small particles act

01:24

like photons and photons have waves we

01:27

can substitute the energy in in this

01:30

equation by H bar Omega and this is kind

01:33

of a leap of Fate but hey that's what

01:35

the brogley did and he came up with the

01:37

correct equation describing the wave of

01:39

a of a small particle like an electron

01:41

so we're doing the same thing here of

01:43

course we didn't do this the first time

01:45

this was done kind of through Char and

01:47

eror so we take the energy in this

01:49

equation which is equal to P ^2 / 2m + V

01:53

and the energy is now going going to be

01:55

replaced by H bar Omega which we derived

01:58

in the previous video we're also going

02:00

to replace the momentum by H bar k which

02:02

we also derived in the previous video

02:05

which means that instead of energy we

02:06

get H bar Omega is equal to and instead

02:10

of P we have p^ S we write h bar^ 2 k^2

02:14

/ 2 M plus v so now our new equation for

02:19

the energy of a particle cannot be

02:21

written as this remember that this is

02:24

only valid for a particle because of

02:27

course we can't have a photon that has

02:30

potential energy like this so we this

02:32

can only be valid for a real particle

02:34

with mass and of course also if mass is

02:37

equal to zero we have this divided by

02:39

zero and that would be

02:41

undefined so now the next thing we're

02:43

going to do is we're going to try to

02:44

come up with a surer equation describing

02:47

this particle in one dimension we know

02:49

that the differential equation has to be

02:52

a linear solution so in other words we

02:55

should be able to write the Sur equation

02:58

as the sum of two line lar equations

03:00

like

03:01

that so what is this equation going to

03:04

look like well they took a big guess

03:06

matter of fact they probably guessed

03:08

quite a number of times try to work it

03:09

out but this was their first guess they

03:12

said well wasn't their first guess was

03:15

probably their 10th or 20 or 30 I guess

03:17

but eventually they came up with this

03:19

concept let's say that this is what the

03:21

Sher equation looks like minus H bar s /

03:25

2 m which is very similar to this except

03:27

they took away the K squ put a negative

03:29

in front of that times the second

03:32

partial derivative of the Sher equation

03:35

with respect to position plus the

03:38

potential energy of the particle times

03:40

the sh equation equals I time the first

03:45

partial derivative of the function now

03:47

you say well how did they get this well

03:50

they did some educated guessing and it

03:52

took them a while to get there so later

03:54

on we'll show you more how that can be

03:58

accomplished so one we assume that to be

04:00

the case then we realize that the old

04:03

familiar equation for a wave equation

04:05

where we have the solution to be a sin

04:07

of KX - Omega t or the equation a * the

04:12

cosine of kx- Omega t no longer works

04:16

because the second part here on the

04:17

right side notice since it's the first

04:19

partial derivative you cannot get this

04:22

this as a solution this no longer works

04:25

however a solution in this form will

04:28

work with this equation and we're going

04:29

going to show you in the third video of

04:31

the of the set of three why this

04:33

equation works but we can say that if we

04:36

assume this to be the differential

04:38

equation of the shary equation here's

04:40

the Sur equation right there we can then

04:42

find the solution in this particular

04:45

format and we know that that one works

04:47

and that's part of reason why they chose

04:49

that equation they said well we're going

04:51

to assume that the equation looks like

04:54

this in its format so therefore in order

04:56

to find the differential equation that

04:59

we can find this as a solution we work

05:01

our ourselves backwards and say this

05:03

must be the format of the equation

05:05

that's really where they that's really

05:07

the way they approached it and of course

05:09

the constants came from the from the

05:11

relationship between photons and small

05:13

particles anyway so they said this is

05:16

what the solution should look like now

05:18

let's go ahead and assume that to be the

05:20

case and let's go to our next video to

05:21

see where they went at that

05:27

point