Quantum Mechanics

What I will assume for this paper is that you already know some basic calculus and linear algebra, if you don't, I suggest the book 𝘘𝘶𝘢𝘯𝘵𝘶𝘮 𝘊𝘰𝘮𝘱𝘶𝘵𝘪𝘯𝘨: 𝘍𝘳𝘰𝘮 𝘤𝘰𝘯𝘤𝘦𝘱𝘵𝘴 𝘵𝘰 𝘤𝘰𝘥𝘦 by Andrew Glassner, and for calculus, I suggest the essence of calculus by blue 1 brown. I would also like to warn you that I am really bad at explaining stuff in depth, I hope I make up for this by explaining literally all I know about quantum mechanics in one paper. Now lets get started!

when talking about a quantum state I mean that as a vector in a Hilbert space, whats a Hilbert space? well its just a space with infinite basis vectors being an option. This property is useful as shown below, (this is a linear combination when is the indexed coefficient and is the corresponding basis)

valid vector (no Hilbert space)

valid vector (Hilbert space)

thats why we need a Hilbert space, as the quantum state vector can and will have infinite discrete sums, just much later on. But now we know that, how do we measure different aspects of the particle (or quantum state)? Well first let me introduce you (a little bit) to eigenvectors, eigenvectors are basically just the vectors that when applied an operator on, don't change direction, Just change magnitude or length, this is rare because most of the time if you apply a matrix on a vector it will change direction. Therefore for the eigenvector of an operator , this would be true:

here, the double use of is confusing, but is the eigenvector of and is the corresponding number that eigenvector is scaled by when operated on by the operator also known as the eigenvectors eigenvalue.

Next lets talk about Hermitian operators, all a hermitian operator does is satisfy this proposition below (lets say the operator was hermitian):

where the dagger here represents the adjoint of the operator, which is all the components of the operator flipped over the diagonal and conjugated (negative imaginary component), as shown below ( represent the components of the matrix):

ok, but why is this useful? well the condition that hermitian operators provide ( ) mean that they form an orthonormal eigenbasis, or a set or orthonormal basis vectors made of an operators eigenvectors. How can we prove this? well first I want to show you that all hermitian operator's eigenvalues are real, as this will be useful later. First lets get the normal operator eigenvector definition:

conjugate it:

since we have two equations:

multiply by on the first one, and on the second:

The only way this statement is true is if , which is only true if doesn't have any imaginary component, which means is real. Therefore the eigenvalues for a hermitian operator are all real. Now we can use this to show that all hermitian operators form an orthonormal eigenbasis, lets look at the operator acting on two different eigenvectors and with two different eigenvalues:

we can also conjugate the nd equation:

inner product trick again!!!:

The only way for this to be a true statement (which it is) is for to equal zero, as then both would equal zero. What does this mean? Well I wont go into too much detail now, but a bra times a ket right after another is also known as the inner product, which acts kind of like a projection tool in some cases, so if the inner product of two vectors is zero, which it is here, it means that those to kets are not parallel or moving in the same direction at all and therefore have to be perpendicular, forming a basis with normalized vectors that are perpendicular to each other (orthagonal) and made of an operators eigenvectors, or in other words, an orthonormal eigenbasis!

so now that we know that hermitian operators form an orthonormal eigenbasis, how can we use this? Well if we have an operator such as the energy operator , its orthonormal eigenbasis represents the states that energy can collapse to, so one basis vector would be energy level zero, one energy level one, and so on, since these are basis vectors you can write any vector a a combination of them. How much the state vector of our particle is made of an energy state vector, like nergy level vector, will give us the likelyhood that that energy state will be collapsed to when the particles energy is measure.

Also the position operator, has a eigenbasis with infinite basis vectors because the particle can be in any position (which of there are infinite). The quantum state vector written in different operators eigenbasis's is shown below by linear combinations (integrals because there are infinite basis vectors):

position eigenbasis - operator

momentum eigenbasis - operator

energy eigenbasis (discrete) - operator

the functions and are coefficient functions as since there are infinite basis vectors, you need infinite coefficients to go with them, so those to wave functions are the functions that output the corresponding coefficients. Integrals are used with continuous basis's because you need continuous or infinite sums. the discrete basis is often used in fields like quantum computing, where you can use something like the spin basis (which only has two basis vectors, corresponding with and ) to model superposition between the and states, also allowing to to make quantum gates to act on those superpositions.

Now, quickly before we touch on the inner product, we need to know what the dot product is, and what kets and bras are, kets are just quantum notation for (usually complex) vectors, while bras are transposed (horizontal) and conjugated (negative imaginary component), so a conjugated complex number would turn from something like to . bras and kets are shown below:

(conjugated and transposed)

This is also written as the adjoint of the ket sometimes, , Now what is the dot product? well its the inner product, but for real numbered vectors, its written like below:

Now I won't go crazy deep into what the inner product is, as I will cover what we can use the inner product for, and anything I say can be done with the inner product can be done with the dot product. But the one thing we do need to know is that when you dot product a vector with itself, it will return the magnitude of the vector squared, because you can rewrite the dot product with any two vectors (here its two of the same) as this: with being the angle between the two vectors, if that angle is zero (and therefore is ) then the dot product equals or or the magnitude of the vector squared. The reason this is important is because to square a complex number (what the inner product is made for) you have to do not just , so this means that to get the magnitude of the complex vector squared, you would have to have the dot product with that change, how this is used is shown below:

Dot Product normal

Inner Product changed to match

expand

turn into vectors

definition of bras and kets

Now we know two ways to write the inner product! The use of kets and bras will be very used in the very scientific and totally professional paper, so I suggest getting used to them.

Now you already know the inner products definition, in both bra - ket and summation form, however I think it would be fun and good practice to derive the inner product summation form for both the discrete case (the one you already know) and the continuous case, as continuous basis's show up often in this paper. we will derive the summations from the bra - ket notation, (I will give a basic explanation of the 's you will see), and are commonly used to represent the average quantum state:

discrete inner product! continuous inner product!

You might not know what those deltas are, but their definitions are pretty simple. All the Kronecker Delta ( ) is, is that it will be if those numbers i and j match, the delta is added because the inner product of basis vectors, if they are the same they are , otherwise they are zero. This means it is when the basis vector on each side matches, or in this case . while the Dirac Delta ( ) is basically just the same thing but for continuous sums, singling out a single value just like the Kronecker Delta does, if you want a better explanation, I suggest quantum sense's video on the dirac delta.

One way you can use the inner product is applying it on a basis vector and a quantum state, generating the magnitude of the projection onto that basis vector, I will just show the discrete case here, but the continuous case is basically the same just with the Dirac delta instead of the Kronecker delta , and there are wave functions involved. This example shows projections onto any general basis vector , obviously this only returns the magnitude of the projection not the projection vector as stated above:

expand

<- the coefficient for

we can also use the inner product for finding out if a vector has a magnitude of , which is required for a valid quantum state, when we derived the inner product this is what we based it off of so i'm glad its able to do this, this is shown below:

(if magnitude is )

some operators can also commute, or not, I'll skip some of the visuals, because I physically can't show that in a text document, but just trust be on the fact that if two operators communicate, this means that (for two random commutable operators and , then they share at least a little bit of their eigenbasis, this means that if you can commute two operators then you can measure the exact quantity of one and the other, but if they do not commute, then you can never know the exact value of both at the same time. It is like this between position and momentum and energy and position.

That is called the Heisenburg uncertainty principle, which I explained really shitty, so if you want an actually good explanation watch one of the may videos on the topic.

Next is the most interesting equation in all of quantum mechanics! thats right! is the Schrödinger equation! Now its hard to explain the Schrödinger equation without visual explanations, so I will have to skip over many things and i will ask you to just accept them, if you can't you can watch more videos on it (I suggest Quantum sense's on youtube). Hang in there!

Lets first imagine what the Schrödinger equation is doing. What it does is describe how time evolution changes a quantum state, it does this actually through a connection to classical physics. As in classical Lagrangian mechanics, a change in time to the Lagrangian is resulted from a change in energy, the Lagrangian is like the quantum state but for classical physics, it holds all information about the object. How this connects and equations shown below (finally I can make equations again):

(this shows that to generate time change you need to look how energy changes in time)

This is one of the things I will ask you to just accept, but trust me this is how the Schrödinger equation is formatted! All the Schrödinger equation is doing is saying the time evolution of a particle is caused by a change in energy (or the Hamiltonian/Energy operator), this assumption comes from experiments and the Lagrangian mechanics equation above. The and is there for probability conservation and unit type consistency respectively. if we expand the (total energy) operator into what its made of then we get or the kinetic energy operator plus the potential energy operator, giving us the full energy operator. This lines up with classical physics perfectly as well.

Now lets say we wanted to read this equation in the position basis (which is the most common choice of basis for the Schrödinger equation), we would do this by using the inner product, as demonstrated earlier, to project the state into the position basis, , but first we will expand the kinetic energy operator into its proper form, which using classical physics as an example, is / :

expand the operator

project onto the basis

move all s over with

(i will explain this step)

I would like to point out that what you might see in articles and textbooks is this definition:

here, the is known as the laplacian, which is just the gradient of a function. so

and

this is basically just the derivative in every direction added to make main derivative, in our case this is just derivative of the second order, since is dimensional, with just the x axis, so all we have to put is , but with wavefunctions of multiple dimensions, like we need to put , so all the s doing is the exact same thing we are doing, but also accounting for higher dimensions which we will use later.

and there you have it! the position-basis Schrödinger equation! To be clear, the derivative turned into a partial derivative because we are now measuring the difference in time only when we also have position as a variable. Now I know i'm not one to explain stuff very deeply, (if you couldn't tell by how horrible my explanations are) but I want to have more space and time devoted to that fourth and final algebraic transformation, so i can explain it better. First we expand all to , we do this because is the same as projecting the quantum state onto the position basis, therefore because projecting a quantum state returns its coefficients in that basis, it should return all the coefficients for the position basis:

projecting the quantum state to the position basis

is a wave function that returns the coefficients

expand the quantum state in the position basis

pull the into the integral

turn into the Dirac delta

use the definition of the Dirac delta

therefore, we can see that should return the wave function , but then why does it return instead? Well, because is (basically) ! the only difference is that the wave function takes account of time as well, so you can look at the position basis coefficients in the future or past. so , as they are both the coefficients of the position basis at the current moment. You can also see that this uses the inner product for projections just like i did earlier when demonstrating what we could use the inner product for, just with a continuous basis and projecting onto a basis instead of a vector.

Next lets talk about the momentum operator we know that the momentum operators eigenbasis forms the momentum basis. But thats not what we are going to use right now, instead we are going to use more Lagrangian mechanics! As a reminder the function is like the quantum state but for classical physics, it holds all information about the object. And last time we only checked what caused a change in time of , which it turned out to be energy, so lets check out the other generators of Lagrangian mechanics:

time evolution (what we used for SE) position change

momentum change Angle change

Now what i'm about to say i cant really prove with text alone, so I suggest quantum senses's series chapter (youtube). Now what we can do is transform these into there quantum versions just like we used the time evolution formula for the Lagrangian to make the Schrödinger equation. all transformations according to the (Lagrangian) are shown below:

time evolution (SE) position change

momentum change angle change

now it is cool to see the Schrödinger equation is actually just one of multiple "changes in something" equations, you just see it more often because a change in time is more significant than the others. But the important part of this we can use the the generator of spacial or position change, which is momentum, , where we can see that if we act on the position basis , we get the equation on the left, we can use this as shown below to find out how the momentum operator acts on a quantum state! (the general idea is integration by parts, but if you don't know that you can just think of it flipping the derivative sign to move to the function)

acting on the quantum state

move to

expand

integration by parts

boundary term vanishes to

move the negative inside

therefore

and projecting that onto results in the classic momentum operator formation:

now that we know more about the momentum operator we can finally explain the final form of the Schrödinger equation in the position basis:

we already described why derivatives turned partial, because we are now dealing with a position variable as well, we already described why the turned into , now we only have two things to tackle, why the is potential energy operator now acting on a position eigenvalue instead of the position operator , and what the fuck happened to ?? well, changed because is an eigenstate of , meaning that when you multiply by , the will turn into 's eigenvalue.

Next, why does / turn into... whatever that is? well we have derived what the momentum operator looks like, so lets write that out and try to understand this:

therefore

so

And there you have it! if you also account for we now having to change the derivative to a partial derivative dues to multiple variables, then you have pretty much completely and utterly derived the position basis Schrödinger equation. Which is shown again below so you can check and revisit any parts you are rusty on:

Below I will try to derive the Schrödinger equation in the momentum basis, both for my own practice and yours, see if you can name everything i'm doing!

there! the transformation was because projected onto the momentum basis is that, proof below:

therefore

therefore

You might have noticed that when you project some eigenstate (like ) onto something involving that eigenstate's operator (like / ), you can just change the operator to the eigenvalue of that eigenstate ( / ). But when you project a eigenstate onto a expression with a operator that is not the eigenstates operator, like onto , you have to transform the operator into what the operator is in relation to the eigenstate's, operator (like / ). I certainly noticed this when making quite a few errors writing the equations down, and now I hope you do to!

Now that we know what quantum operators are, the inner product, what quantum states are, how to find quantum operators, and the Schrödinger equation, I believe its time to solve the it! In concept it sounds easy, if we can take the position basis Schrödinger equation:

and solve for , we can then use to find :

and since is the quantum state at a certain time, we can find the quantum state in the future, or predict what it will do. So is the current state and is the quantum state something like seconds in the future. Easy!

But actually it's not that simple... The main problem is solving for , as differential equations are by nature hard to solve, made even harder with partial and second order derivatives. So what we do instead is transform that position basis Schrödinger equation into the TISE, or Time Independent Schrödinger Equation. This (as you might have guessed) simplifies the Schrödinger equation to a more manageable form based on the assumption that the potential energy operator is not dependent on time.

The first thing the TISE calls for is the separations of variables, aka turning into two functions, shown below:

this makes no partial derivatives needed

(I would like to point out is usually or in professional work but I don't want confusion with the momentum wave function or position wave function so I use ) since we don't need partial derivatives, lets convert the partial derivatives into terms of and , a function that only considers time, and a function that only considers position, some rules do need to be met to make this allowed but I am not going to explain them because separation of variables is one of the things I don't understand very well. the conversion steps are shown below (try to logic your way through the steps):

substitute:

Next, divide by :

Now, what I am about to say is quite weird, but makes sense if you think about it. Since the left side only depends on time, and the right side only depends on position, this means that both sides are constant. This is because if you change the right side doesn't change (because the right only depends on ) so the left side (where you changed ) shouldn't either, same with changing , the left side doesn't change so the right shouldn't either. This means that both halves of the equation are constant, even if you change and . Since both halves are constant, this means that both sides equal some constant number , because:

(since both sides are constant and equal)

Now that we know this, lets solve the left hand side of the equation for , these steps are shown and explained below:

already proven above

move

(i is negative because )

move over

integrate

expand left to make sense

integrate (actually)

exponentiate (rid of )

exponent rule

is some constant

so there are many things i would love to explain in more detail on why we can move the numbers in that way and how we are able to do it, but there are too many things to talk about and it would go off topic, if you want an explanation on the TISE other than this paper, I would suggest my favorite learning technique, find a good youtube video, like https://www.youtube.com/watch?v=Dt_VKsSggAo, and ask some AI or expert on the topic at any points you are confused on.

Now solving for is a lot easier, as all that is needed is a little multiplication to turn it into the TISE, steps are shown below:

multiply each side by

And there you have it! the TISE in full glory, How we can use this TISE is we can solve for and then use (and our definition of ) to get , which can be used to derive our quantum state , as is just the quantum state in the position basis. How we use gamma to get and then get is shown below:

use definition of

use definition of

use definition of

Notice how the constant of integration is gone from the definition , the was removed because it was a constant of integration, and the normalization rule of any wave function (like ) forces it to be a certain value, aka if isn't set to a certain value the normalization rule is false, which is not true, so we set to a fixed value where the normalization rule fits, and then just put built into the definition, so we don't have to write it anymore, but its still there:

...Now I have lied to you slightly, in many practical uses is not true, this is because is an eigenstate of the eigenbasis the Hamiltonian, or energy operator creates. So for that definition of to be true, the energy state would have to be collapsed onto one , which will not be the case of many many particles. How do I know is an eigenvector of the energy eigenbasis? well look back at the TISE:

the TISE

left is just

since is a constant, this means that applying the Hamiltonian operator to scales it by a constant , you know what that means gamma is? an eigenvector / eigenstate of the Hamiltonian operator! This is because... well thats literally the definition of an eigenvector of an operator, applying that operator to that state only scales that state by a factor. This implies really cool things, because if you measure energy, energy collapses onto one eigenvalue of the energy eigenbasis, but what it also does is collapse position into one or position wave function, restricting where the particle can be, but back on topic.

But if isn's true — in most cases — what is? Well since is not usually just value, as that would imply that the particle has collapsed into eigenstate , all we have to do is write as instead of a single value, as a full eigenbasis, which means we can use our definition from the literal beginning of this paper to expand to instead be a summation, using coefficients and distributing the to each component, which we can then use to derive the quantum state:

I would also like to add that the TISE only happens when the Hamiltonian or energy operator doesn't depend on time, aka the energy of a particle doesn't change over time, now in real life this often isn't true, but most of the time for simulations the TISE is just fine for simulating a particle in a perfectly static box, or potential energy well.

Ok. Now that we finally fully understand the TISE, what its used for, what every component means and why the is often inaccurate, replacing it with a new term, I think its darn time we finally figure out how to solve the TISE.

Now what you may have noticed is that the wave function we are using is , only taking in a d space , and trust me we will learn how to solve the d TISE with later, but first we will learn how to solve the d TISE, and use what we learned to solve the d version.

the first thing we need to establish is how we set up the d walls, so we can simulate the particle in a d box, how we can do this is say that the particle would need potential energy in those regions, aka the particle can't be there. Next we can say the particle has potential energy in the box, as nothing is acting on the particle, meaning we can remove when solving in the box. Using this information, we can simplify the TISE as shown below when in the walls:

(since is )

now, what do we do? well is a constant, and so is , so lets move to the right with to single out the second derivative:

where

The simplification of might seem odd, but it will help us in recognizing a pattern. Now we just have to find what function has a second derivative that results in itself times a negative constant squared. This isn't that hard as there is a function that does exactly that, the function, the function returns a second derivative of , but if we add a constant like , can you guess what we get? that's right, that constant (in this case ) negative squared:

, so if we put there, we get this:

as you can see, it perfectly matches our equation with a substituted :

so must be , there is a second solution, its , however this cant work for a d box with boundaries at and . why? oh boy this is gonna take a while. So the equation we just derived from the TISE when we are measuring a d box is a second order differential equation (based off the fact that its highest derivative is second derivative), and the second order differential equations are linear (in general) this includes the TISE! Why is this important? Well first I would like to say that specifically in this case our linear operator is / , which is operating on gamma, where gamma is then distributed to the derivative and , this operator is made by taking our equation and moving everything to the left as shown below:

move stuff over

group

is our operator

Now we know that, we need to know that linear operators follow rules, lets call our operator for now, the two rules are shown below:

Why is this important? Well we know from our deriving of our operator that any valid function inputed into the operator must equal zero, or the equation wouldn't be true! But we do know that (and the second solution ) are both solutions to this equation, as shown below:

But what if we want solution that covers all basis's? well its your lucky day! as proven below, we can linearly combine these two solutions and still get a correct answer:

tada! so now we know that is also a solution to this equation! So... what about the whole reason I went into this rabbit hole? Why don't we see or use ? Well this is a general solution to the d TISE, but what if we add those impenetrable potential energy walls? well thats exactly what and are meant for, as they are set by boundary conditions. Now we know the particle is not in the walls, aka the wave function is zero there, but what about at the walls? well there it must be zero to, because if the function you pick does not naturally fall down to zero at the walls then you have an infinite slope at the walls, which would break our derivatives of .

Using this, we can say what functions we can use at (the first wall) and (the second wall), first lets see what we can prove if we say the at the first wall the function has to be zero:

this means that B must equal zero!

Now what about the other side of the box, the wall at does that force A to become a specific value? well, lets see shall we:

since cant be zero or every output would be zero

since is only zero if is a multiple of , such as , we can say:

where n , , ,

divide by on both sides

substitue in :

Now this realization is really cool, as it reveals the principle of quantization, or the fact that a particle can only have discrete energy values, not a continuous set. This proves this because remember how is related to energy in being equal to ? well that means that is related to energy, obviously, as is an energy eigenvalue, but out previous definition also says that is discrete, meaning that the energy of any particle is in discrete values, with no energy levels of a particle allowed between them. This is probably a loose definition but I don't care enough to give a proper one. Now lets get back on topic!

This new definition of is cool, but what about ? we didn't specify a specific value for it and it didn't seem to force to be a certain value. And you would be right, but we can force to be a value, and it comes from normalization, normalization is just the rule that all coefficients magnitude squared add up must be equal to one, this is because the magnitude squared of a coefficient is the probability of measuring the particle in that eigenvector. Since probabilities must add up to , or , then below is true:

Now I am going to do the math but not do a lot or any explaining at all, so be wary:

use trig definition:

split the integral:

solve left side

solve right side:

substitute into full equation:

(remember?)

and

Finally! that was a lot of continuous math I am sorry, and there are many points where you might have been confused, but we found a value for finally! that means we can finally know what the full is for the d TISE when you have boundary conditions at and , I will show how to go from general solution to our solution now that we know what each step is, try to remember each step!:

expand

expand

And there we have it! we have a fully working function for , however this is the position wave function at a single point in time, what about that Time evolution the Schrödinger equation promised? Well... do you remember ? we can finally use our definition from ages ago to find the dreaded ! We do have to make a few edits though:

you can see n turned to (because they do the exact same things) and we changed to because there are different energy eigenvalues for each energy eigenvector . I would also like to point out that we are summing to infinity, this is because theres technically nothing stopping you from just adding more and more energy to the particle forever, even though energy is discrete and not continuous (this is why we use summation instead of integral, it sums infinitely but is discrete). But in practice, the coefficients and therefore the probability of a high energy state goes down very fast, so summing to is good enough for pretty much everything:

Now thats how we find the time evolution of a particle in a d box, but what about a d box/square? well thats actually pretty easy, first lets look what the d TISE looks like and the d TISE looks like:

d TISE: (assuming no in the box)

d TSIE:

Now we can turn the d version into d TISE's, how we do this is the same thing we did to get the TISE out of the normal Schrödinger equation. The separation of variables! We can turn into and , so , we use this to get two different d TISE's below:

divide by

distribute

the only way for sides (both depending on different variables) added together can equal a constant , is for those two sides to also be constants, and . Using these new constants, we can split this equation into two d TISE's and solve them!

multiply both sides by

d TISE!!

multiply both sides by

d TISE!!

Now, lets say that the based TISE's boundary positions are and , while the based TISE's boundary conditions are and . Using this we already know the solution to these d TISE's, this is because we already know what the solution to the TISE looks like! so we can just substitute in our boundary conditions and bam! we have a solution:

solution for based TISE:

solution for based TISE:

we can then combine the two square root values, and find , then using that to find :

combine square root values:

then use to find :

We used a double summation because we need to allow the and coordinates to have different energy values, creating different eigenstates, like , , low x energy and high y energy. But there you have it (once you understand what ive just said) a particle moving through time and space in a stationary box! now you can probably guessed based on the pattern how the d form looks:

d :

d :

d :

therefore, (for the people that just want an absurd equation) this is how to find the quantum state itself in d space, which you don't need to find, as is the quantum state, just in the position basis, aka is just :

or simplified:

tada! all the position and quantum state information and equations you will ever need! If you want to find out how to solve the Schrödinger equation with the hamiltonian depending on time (aka the particle is allowed to change energy levels through time)... I'm not gonna tell you! Go find it yourself. Ive done enough math, and you probably have to, thanks for reading! and I hoped you understood at least a little about what Im saying, you can use the d, d, or d position-time equations to simulate a particle or atom in a perfectly static box, with the x boundaries being to , y boundaries being to , and z boundaries being to , this is what I am planning to do, however I don't know any d software thats good for this. Problems, problems...

Oh! if you where wondering, this doesn't just limit to d d or d, you can go as high as you want, therefore I present the totally stupid big dimensional equation for finding the position wave function evolution over time of a dimensional particle and projecting it to make the quantum state vector: