What is the Significance of Schrodinger's Wave Equation in Quantum Mechanics?

[Priyadarshini]

Why would one need to use the Schrödinger's wave equation? What does it give us? I understand that we solve for psi and stuff, but graphically what does it give us, in practical and not theoretical terms? What exactly is a wave function? After solving for psi with the help of the equation, we get multiple psi values. How do we know which are the "Eigen wave values"? Also, what do you even mean by a "true, meaningful"(this is how the video I watched describes Eigen wave values) psi value? Psi^2 is the probability function which tells us the probable position of the electron within the atom, why can't we just use the Heisenberg's uncertainty principle to find this instead of solving for psi using this longer, more complicated formula?
Thanks in advance!

 

[mfb]

Why would one need to use the Schrödinger's wave equation?

How else can you do anything? Every other model is based on this (or the more general Dirac equation or quantum field theory, but let's ignore that here) or equivalent formulas.

What does it give us?

It allows to calculate electron orbitals and energies, it allows to predict chemical bonds and so on. Basically everything.

What exactly is a wave function?

That's philosophy. For physics: a very useful tool.

How do we know which are the "Eigen wave values"?

Given by the mathematics. Some satisfy the equation for eigenstates, some do not.

why can't we just use the Heisenberg's uncertainty principle to find this

The uncertainty principle just tells you that the position cannot be a single point (and gives a lower limit on the uncertainty), it does not tell you how likely it is to measure the particle at some specific point.

 

[DEvens]

Your shotgun approach is a little off-putting. Imagine somebody who you are trying to teach to drive a car who kept asking you questions about "why can't we do this?" and "what is the purpose of that?" and "shouldn't this go over here?" You keep telling him to put his hands on the steering wheel, and he keeps bothering you about "What is in the glove compartment? We can't drive the car without the registration."

S's equation is a particular way to deal with quantum mechanics from a given point of view. You can show, with more work than I am prepared to do just at this moment in this forum, that the S.E. is equivalent to a particular limit of the Feynman path integral. That is, solving the S.E. is a way of determining the wave function of a quantum particle in an energy potential. In quantum mechanics, the wave function contains all the information about a particle that exists. You use the wave function to predict the outcome of experiments, to determine energy levels, to determine the probability of interactions, to determine half life to decay, etc.

I don't know what video you were watching. And I am a little hazy on the two words you have taken out of context. And the term "Eigen wave value" sounds like you have mixed up Eigen vector and Eigen value, two different but related things.

Possibly you could get an introductory quantum mechanics textbook and do some reading?

 

[Priyadarshini]

Your shotgun approach is a little off-putting. Imagine somebody who you are trying to teach to drive a car who kept asking you questions about "why can't we do this?" and "what is the purpose of that?" and "shouldn't this go over here?" You keep telling him to put his hands on the steering wheel, and he keeps bothering you about "What is in the glove compartment? We can't drive the car without the registration."

S's equation is a particular way to deal with quantum mechanics from a given point of view. You can show, with more work than I am prepared to do just at this moment in this forum, that the S.E. is equivalent to a particular limit of the Feynman path integral. That is, solving the S.E. is a way of determining the wave function of a quantum particle in an energy potential. In quantum mechanics, the wave function contains all the information about a particle that exists. You use the wave function to predict the outcome of experiments, to determine energy levels, to determine the probability of interactions, to determine half life to decay, etc.

I don't know what video you were watching. And I am a little hazy on the two words you have taken out of context. And the term "Eigen wave value" sounds like you have mixed up Eigen vector and Eigen value, two different but related things.

Possibly you could get an introductory quantum mechanics textbook and do some reading?

Sorry about the shotgun approach, I tend to get a little worked up when I don't get things.
Oh, okay. I haven't done integration yet.
-- At around 7 minutes, he starts talking about "true, meaningful" values.
What books would you recommend for me? (I'm still in high school)
Thanks for your help!

 

[Priyadarshini]

How else can you do anything? Every other model is based on this (or the more general Dirac equation or quantum field theory, but let's ignore that here) or equivalent formulas.
It allows to calculate electron orbitals and energies, it allows to predict chemical bonds and so on. Basically everything.
That's philosophy. For physics: a very useful tool.
Given by the mathematics. Some satisfy the equation for eigenstates, some do not.
The uncertainty principle just tells you that the position cannot be a single point (and gives a lower limit on the uncertainty), it does not tell you how likely it is to measure the particle at some specific point.

What is the equation for eigenstates?
Oh! I see why the uncertainty principle can't be used here! Thanks!

 

[DEvens]

.
What books would you recommend for me? (I'm still in high school)

Before you can usefully study QM you need calculus. And some matrix algebra would not hurt you. For example, the video has things like $\frac{d^2\psi}{dx^2}$. Until expressions like that are at least meaningful to you, you will find QM fairly opaque.

Your high school may have calculus classes. If it does, go take them. Try to find some high school level calculus textbooks. It has been way too long since high school for me to suggest a good calculus text. There is a "for dummies" book, and one from Schaum's. They might be ok at high school level.

 

[Priyadarshini]

Before you can usefully study QM you need calculus. And some matrix algebra would not hurt you. For example, the video has things like $\frac{d^2\psi}{dx^2}$. Until expressions like that are at least meaningful to you, you will find QM fairly opaque.

Your high school may have calculus classes. If it does, go take them. Try to find some high school level calculus textbooks. It has been way too long since high school for me to suggest a good calculus text. There is a "for dummies" book, and one from Schaum's. They might be ok at high school level.

We have calculus on our A level syllabus at school. We have only covered the derivatives portion.
Thanks for the book suggestions, I'll have a look!

 

[DEvens]

Heh. The guy's accent in the video is very difficult to get past. It reminds me of a prof from my grad school days. That prof warned his audience that he may mispronounce some things because he worked mostly in isolation. So when he talked about particles having tragic Tories, we should not be too upset.

This guy keeps talking about the skro-din-jers equation. It really jars me out of trying to understand what he is saying.

 

[Priyadarshini]

Heh. The guy's accent in the video is very difficult to get past. It reminds me of a prof from my grad school days. That prof warned his audience that he may mispronounce some things because he worked mostly in isolation. So when he talked about particles having tragic Tories, we should not be too upset.

This guy keeps talking about the skro-din-jers equation. It really jars me out of trying to understand what he is saying.

Yeah, I know. But you get used to it after a while. And since I have been listening to all his videos on this chapter, I finally get what he is saying!

 

[mfb]

What is the equation for eigenstates?

You'll need calculus and linear algebra to understand it fully, but the basic idea:
You apply some "operation" on a function (like taking the derivative) and get another function. If the new function is a multiple of the old function, this function is an eigenstate (=eigenfunction) for this "operation" (operator).

The exponential function is an eigenfunction for the derivative operator, because its derivative is 1 times the original function. That specific example cannot be applied to quantum mechanics and has some other issues but it should give the idea.

 

[Priyadarshini]

You'll need calculus and linear algebra to understand it fully, but the basic idea:
You apply some "operation" on a function (like taking the derivative) and get another function. If the new function is a multiple of the old function, this function is an eigenstate (=eigenfunction) for this "operation" (operator).

The exponential function is an eigenfunction for the derivative operator, because its derivative is 1 times the original function. That specific example cannot be applied to quantum mechanics and has some other issues but it should give the idea.

Thanks!