Help with the Schrödinger Equation.

[96hicksy]

So I've been trying to understand this equation for a long time, about half a year - I don't know someone who could teach me this either due to the fact that I am only 15 and going into my final year in secondary school after August and according to most colleges (UK college, not American) in my area, they don't teach how to understand/use this equation, and due to my lack of knowledge - I can't make heads or tails of it, bar the odd exception.

Not knowing how to do the equation seriously bugs me as I constantly require to seek information so I thought i'd come back to Physics Forums in a bid to help.

Can anyone on the site help me to understand even the simplest version of the Schrödinger equation? It'd help me out a lot. If its not much trouble either, an explanation of the Wave Function would be amazing - or recommended books.

Thanks, Ben. :)

 

[Mark M]

In simplest terms, a wavefunction is a mathematical object that describes the probabilities of observing the a particle (or group of particles) to be located to be located at a specific position. This is crucial, because it make an important statement - the most you can speak about a particle's position in quantum mechanics prior to the actual observation is the probability that it is in a certain position. The wavefunction encodes this information about the particle. The Schrodinger equation governs the behavior of a wavefunction through time. It's a wave equation, so the wavefunction evolves in the same manner a classical wave does - it spreads out. If I have a general idea where a particle is, it's position is localized - it may have a 95% probability of being here, and a total of 5% for the nearby surrounding areas. However, similar to the way waves spread out, this probability distribution will spread out over time, and I will be uncertain about it's position.

The exact way to calculate the way a wavefunction evolves is with the Schrodinger equation. Actually deriving solutions is not a simple affair, but you can still get a general idea of what the content of the equation is by looking at it's parts. Here is the time dependent equation in one dimension:

$$i \hbar \frac { \partial \psi} {\partial t} = H \psi$$

First, look at the 'i'. That's the imaginary number, the square root of negative one. It has an important role in any wave equation, but we aren't interested in it at the moment. More important is $\hbar$ (pronounced 'h-bar'), Planck's constant (well, it's the Reduced Planck constant, the normal Planck constant divided by $2 \pi$), it a fundamental constant of nature. It essentially determines the scale at which quantum mechanics becomes relevant. Namely, if you multiply it by the frequency of a ray of light, you get the energy of a photon. It is also the fundamental unit of an important concept in quantum mechanics called spin.

But, both of those are constants. The next part of the left side is the time derivative of the wavefunction. A time derivative is an expression that represents how a quantity change through time. If I take the time derivative of position, I get velocity. If I take the time derivative of velocity, I get the acceleration. So, this term describes how the wavefunction changes in time.

On the right we see the wavefunction again. The important part is the 'H', which is called the Hamiltonian operator. It essentially represents the energy of the particle. It's an operator because it 'operates' on the state of the particle (the wavefunction).

So, now we can see the underlying purpose of the Schrodinger equation. The energy values of the particle, through the Hamiltonian, generate the time evolution of the wavefunction.

Was that helpful?

 

[gildomar]

Can you clarify exactly what you mean by the "simplest version"? Because in the broadest brushstrokes, as far as I know, the equation can just be summed up as Total Energy = Kinetic Energy + Potential Energy. Most other ways of writing the equation though require an understanding of partial differential equations in order to work with it. And I'm not sure how much math you've had, but that area of math isn't covered until advanced calculus.
And what quite are you looking for about an explanation of the Wave Equation: how it came to be, the physics of it, the implications of it, etc. ?

 

[96hicksy]

$$i \hbar \frac { \partial \psi} {\partial t} = H \psi$$

Would you be able to give me an example of this equation in use?

 

[Simon Bridge]

You need to know about complex numbers, probability density functions, and differential equations.

The Schödinger equation is how you find valid wave-functions.

The wave function is a function that can have complex values ... when you pre-multiply it by it's complex conjugate you get a function of real numbers ... which is the probability density function for whatever it was the wavefunction was for. eg. if the wavefunction was for position x then $\int_a^b \psi^\star \psi \cdot dx$ is the probability of finding the particle in the regeon a < x < b and $\int_{-\infty}^\infty \psi^\star \psi \cdot dx = 1$

You should realize that it can take college students roughly half a year to figure it out enough just to use it confidently so don't fret. Google for "introduction to schodinger equation" and you'll be spoiled for choice - pick through them (there are some in video for eg) until you find something that makes sense to you.

If you have any specific questions, you can ask here.

[oh look we all leapt in :) ... there should be something to go on with from all the above]

 

[Mark M]

$$i \hbar \frac { \partial \psi} {\partial t} = H \psi$$

Would you be able to give me an example of this equation in use?

I can, but as I said, it isn't simple. How comfortable are you with the trigonometric functions? If you aren't familiar with trig, do a search for an explanation of what the trig functions are and how they're related. You absolutely need to understand these to understand any wave equation.

 

[unchained1978]

The Schrodinger equation is often presented to be more mysterious that it should be. If you're familiar with the language of operators and linear algebra, than it's actually pretty straightforward. If you don't have a solid understanding of some of the basic results of linear algebra and complex numbers, then I'm sure it is confusing. The Schrodinger equation in this context is simply a description of the energy of a system. Recall from classical mechanics, the Hamiltonian is defined as $H=\sum_{i}p_{i}\dot{q}_{i}-\mathcal{L}$, and in the event that there is no time dependence of the coordinates, this is simply the total energy of the system. $H=p^{2}/2m+V(r)=E$. Now simply promote the constituents of this expression to operators with their corresponding eigenvalues and you have the Schrodinger equation. $H|\psi\rangle=E|\psi\rangle$ The wavefunction represents the probability amplitude for a system to be found in a certain configuration. A good way to gain some intuition for it would be to examine a few solutions to some simple systems, such as the free particle, the particle in a box, infinite square well and so forth.

 

[96hicksy]

I don't have a basic understanding of linear algebra or complex numbers - any idea on where to start?

 

[Mark M]

I don't have a basic understanding of linear algebra or complex numbers - any idea on where to start?

You know regular algebra, right? The best place for linear algebra is MIT's course by Gilbert Strang. Do a search on YouTube for MIT 18.06 and you should find it.

For complex numbers, you should find a good explanation by searching something like 'complex numbers explained'. The core idea isn't too difficult to grasp.

 

[96hicksy]

I've been reading up on complex numbers and they don't seem too difficult to grasp - ill focus more on them later. I'll also watch the video later and then come back to here, thanks for the help!

 

[Mark M]

I've been reading up on complex numbers and they don't seem too difficult to grasp - ill focus more on them later. I'll also watch the video later and then come back to here, thanks for the help!

Well, it isn't a video, it's a long series of videos, all of which are about 40 minute lectures. It'll take quite a while to get through them all, but I recommend it if you're intent on understanding the mathematics of quantum mechanics.

 

[96hicksy]

I watched a little of the lectures last night (UK time) but I did not have enough time to complete one, however I searched the internet for a pdf on linear algebra, how is this? http://joshua.smcvt.edu/linearalgebra/book.pdf

 

[airer]

I think it is ok for an English high school student (average) but too low for a french student. If you want to be genius in Maths, I will advice you to check french books that is if you know a little bit of french. Our English system sucks when it comes to maths.

Good luck kido!

 

[unchained1978]

It seems good enough for an introduction. Pay careful attention to the section dedicated to vector spaces, as this concept is generalized in the context of quantum mechanics and is hugely important. How comfortable are you with classical mechanics? It seems to me you may be jumping ahead too far, you should have a strong understanding of Hamiltonian mechanics if you're to make the leap to quantum.

 

[mikeph]

You don't need to learn all linear algebra, just what a matrix is, how to multiply matrices and vectors, and, from that, to understand what eigenvectors and eigenvalues are. Doing a load of exercises should be enough.

And regarding the Schrodinger equation, definitely start with the time-independent version HΨ=EΨ, where H is the hamiltonian operator of the system. Once you learn eigenvalues and eigenvectors, that equation should begin to look familiar.