**State**

A *state* is something that a *system* is in. The system is where I perform my measurement; the state is the result of that measurement.

When I perform one kind of a measurement, and then I perform another kind of a measurement, the two results are correlated. In particular, the first result may determine the probability that the second measurement will yield a specific value, i.e., that the system will be in a specific state with respect to the second measurement, as opposed to some other state.

The abstract symbol for a state \(x\) is \(|x\rangle\). This is just a label; it is not a number.

The symbol for the transition from state \(x\) to state \(y\) is \(\langle y|x\rangle\). This, actually, is a number!

**Amplitude**

Specifically, it is a *complex number* called the *amplitude*. The reason why it is a complex number is experimental: we found that if there are two possible ways for a system to reach state \(y\) starting from state \(x\), it is not the probabilities, but these complex numbers that will need to be summed:

**Probability**

The actual probability that the system in state \(x\) will also be in state \(y\) is computed as the square of the absolute value of the complex number:

**Base states**

It is assumed as an axiom that any state can be expressed as a sum of * base states*:

What we know about the base states is that they are *orthogonal*:

\[\langle i|j\rangle=\delta_{ij}.\]

**State vector**

The contribution of each base state \(|i\rangle\) to \(|x\rangle\) is characterized by \(\langle i|x\rangle\), which is just a complex number. The state \(|x\rangle\) can, therefore, be viewed as a *vector* that is expressed in terms of *base vectors* \(| i\rangle\) in some complex vector space.

The transition amplitude from state \(x\) to state \(y\) can be expressed through a set of base states as:

The probability that a system in state \(x\) is in state \(x\) is unity:

\[\langle x|x\rangle=\sum\limits_i\langle x|i\rangle\langle i|x\rangle=1.\]

The probability that a system in state \(x\) is found in *some* base state is also unity:

\[\sum\limits_i|\langle i|x\rangle|^2=\sum\limits_i\langle i|x\rangle\langle i|x\rangle^\star=1.\]

From this one can see that

\[\langle i|x\rangle=\langle x|i\rangle^\star.\]

And since any state \(y\) can be a base state in some set of base states, it is true in general that

\[\langle y|x\rangle=\langle x|y\rangle^\star.\]

**Operators**

When you *do* something to a system, you *change* its state. This is expressed by an *operator* acting on that state:

\[|y\rangle=\hat A|x\rangle.\]

This is *defined* to mean the following:

\[\hat A|x\rangle=\sum\limits_{ij}|i\rangle\langle i|\hat A|j\rangle\langle j|x\rangle,\]

which means that \(\hat A\) is just a collection of matrix elements \(A_{ij}\), expressed with respect to some set of base states.

**Expectation value**

A measurement may be expressed in the form of an operator. If this is the case, the average (expectation) value of that measurement can be expressed as:

\[A_\mathrm{av}=\langle x|\hat A|x\rangle,\]

which really is just shorthand for

\[A_\mathrm{av}=\sum\limits_{ij}\langle x|i\rangle\langle i|\hat A|j\rangle\langle j|x\rangle.\]

**Wave function**

What if there is an infinite number of states? For instance, a particle's position \(l\) along a line may be expressed in terms of the set of individual positions as base states. But there is an infinite number of such positions possible. Thus, our sum

\[|l\rangle=\sum\limits_x|x\rangle\langle x|l\rangle\]

becomes instead the integral

\[|l\rangle=\int|x\rangle\langle x|l\rangle dx.\]

This equation has little practical meaning since \(|x\rangle\) is just an abstract symbol. However, the probability that a system in state \(l\) is later found in state \(k\), previously expressed as the sum (1):

\[\langle k|l\rangle=\sum\limits_x\langle k|x\rangle\langle x|l\rangle.\]

is now the integral

\[\langle k|l\rangle=\int\langle k|x\rangle\langle x|l\rangle dx.\]

Both \(\langle k|x\rangle\) and \(\langle x|l\rangle\) are just complex numbers; complex-valued functions, in fact, of the continuous variable \(x\):

\[\langle k|x\rangle=\langle x|k\rangle^\star=\psi^\star(x),\]

\[\langle l|x\rangle=\psi(x).\]

These functions are called *wave functions* mainly because they typically appear in the form of periodic complex-valued functions. With their help, the transitional probability can now be expressed as:

\[P(\phi\rightarrow\psi)=\int\phi^\star(x)\psi(x)dx,\]

and the expectation value of an operator can be written as

**Algebraic operator**

In this context, \(\hat A\) no longer works as a matrix operator converting a state vector into another state vector, but as an algebraic operator converting a wave function into another wave function. How the matrix operator, expressed in terms of states and amplitudes, and the algebraic operator, expressed usually as a differential operator, relate to each other is another question!

**Position operator**

If we know the probability \(P(x)\) that a particle will be at position \(x\), we can compute the average position of the particle after many measurements as follows:

\[\bar x=\int xP(x)dx.\]

But this is the same as

\[\int x\phi^\star(x)\phi(x)dx=\int\phi^\star(x)x\phi(x)dx,\]

which is formally identical to the expectation value (2) for a measurement that can be expressed in the form of an operator \(\hat x\). In other words, \(\hat x\) can be viewed as the *position operator*. When the base states are positions, the position operator is just a multiplication of the wave function by \(x\).

**Momentum operator**

The same computation can be performed for the momentum, using as base states states of definite momentum:

\[\int p\phi^\star(p)\phi(p)dp=\int\phi^\star(p)p\phi(p)dp.\]

Question is, can the momentum operator be expressed in terms of base states of position?

The amplitude of a system, which is in state \(\beta\), to be found in a state of definite momentum \(p\), is just the definite integral

\[\int\limits_{-\infty}^\infty\langle p|x\rangle\langle x|\beta\rangle dx.\]

The relationship between position and momentum, specifically the amplitude for a particle to be found at position \(x\) after it has been measured to have momentum \(p\) is *assumed* to be

So our integral becomes

Now let's use, as \(|\beta\rangle\), the state \(\hat p|k\rangle\) (it can be any state after all) where \(\langle x|k\rangle=\phi(x)\), and the \(k\)'s are assumed to be states of definite momentum. This way, our earlier expression becomes the expectation value of the momentum:

\[\int\limits_{-\infty}^\infty\langle p|x\rangle\langle x|\beta\rangle dx=\int\limits_{-\infty}^\infty\langle p|x\rangle\langle x|\hat p|k\rangle dx=\int\limits_{-\infty}^\infty\langle p|x\rangle p\langle x|k\rangle dx.\]

Then \(\langle x|\beta\rangle\) is just \(p\langle x|k\rangle=p\phi(x)\), and:

\[\langle p|\beta\rangle=\int\limits_{-\infty}^\infty e^{-ipx/\hbar}p\phi(x)dx.\]

The integral can be computed by observing that \(de^{-ipx/\hbar}/dx=(-i/\hbar)pe^{-ipx/hbar}\), integrating in parts, and assuming that \(\phi(x)=0\) when \(x=\pm\infty\):

\[\langle p|\beta\rangle=\frac{\hbar}{i}\int\limits_{-\infty}^\infty e^{-ipx/\hbar}\frac{\partial\phi}{\partial x}dx,\]

so, from (3):

\[\langle x|\beta\rangle=\frac{\hbar}{i}\frac{\partial\phi}{\partial x},\]

and we now have an expression for the momentum operator \(\hat p\):

\[\hat p=\frac{\hbar}{i}\frac{\partial}{\partial x}.\]

**Time displacement**

How does a system evolve over time? Let's consider the *time displacement operator* \(\hat U(t_1,t_2)\):

\[\langle\chi|\hat U(t_1,t_2)|\phi\rangle.\]

**S-matrix**

When \(t_1\rightarrow-\infty\) and \(t_2\rightarrow+\infty\), we call \(\hat U(t_1,t_2)\) the *S*-matrix.

Making \(t_1=t\) and \(t_2=t+\Delta t\), observing that when \(\Delta t=0\), \(U_{ij}\) (in some coordinate representation) must be \(\delta_{ij}\), and *assuming* that for small \(\Delta t\), the change in \(\phi\) will be linear, we get:

\[U_{ij}=\delta_{ij}-\frac{i}{\hbar}H_{ij}(t)\Delta t.\]

(the factor \(-i/\hbar\) is introduced for reasons of convenience.)

In other words, the *difference* between the wave function of the two states can be expressed as:

\[\phi'-\phi=-\frac{i}{\hbar}\Delta t\bar H\phi,\]

or, dividing by \(\Delta t\) and recognizing the left-hand side as a time differential:

\[i\hbar\frac{\partial\phi}{\partial t}=\bar H\phi.\]

**Schrödinger equation**

Schrödinger, that kind chap, then just decided to use in place of \(\hat H\) an operator that he concocted up on the basis of the *classical* expression for energy:

\[E=\frac{p^2}{2m}+V.\]

His equation:

describes the wave function of a particle moving in a potential field \( V\).

**A crucial thought is that the Schrödinger equation is not as fundamental as you might have been led to believe. Indeed, there's no single Schrödinger equation; the actual equation of a system depends on the characteristics of that system, and is often derived heuristically, through the process of operator substitution.**

**Operator substitutions**

One result is a "rule of thumb": substitution rules that are used to derive quantum operators from the classical quantities of momentum, energy, and position:

\[\hat p\rightarrow\frac{\hbar}{i}\frac{\partial}{\partial x},\]

\[\hat H\rightarrow i\hbar\frac{\partial}{\partial t},\]

\[\hat x\rightarrow x.\]

**Commutativity**

The operators \(\hat x\) and \(\hat p\) do not commute:

\[(\hat x\circ\hat p)\phi=x\frac{\hbar}{i}\frac{\partial\phi}{\partial x},\]

\[(\hat p\circ\hat x)\phi=\frac{\hbar}{i}\frac{\partial(x\phi)}{\partial x}=\frac{\hbar}{i}\frac{\partial x}{\partial x}\phi+\frac{\hbar}{i}x\frac{\partial\phi}{\partial x},\]

\[(\hat p\circ\hat x-\hat x\circ\hat p)\phi=\frac{\hbar}{i}\phi.\]

**Probability Current**

A simple manipulation of the Schrödinger equation—multiplying on the left by \(\phi^\star\), multiplying the equation's complex conjugate on the left by \(\phi\), and subtracting one from the other—can lead to the continuity equation:

\[\phi^\star\left(\frac{\hbar^2}{2m}\nabla^2\phi+V\phi-i\hbar\frac{\partial\phi}{\partial t}\right)-\phi\left(\frac{-\hbar^2}{2m}\nabla^2\phi^\star+V\phi^\star+i\hbar\frac{\partial\phi^\star}{\partial t}\right)\]

\[=\frac{-\hbar^2}{2m}(\phi^\star\nabla^2\phi+\nabla\phi^\star\nabla\phi-\nabla\phi\nabla\phi^\star-\phi\nabla^2\phi^\star)-i\hbar\left(\phi^\star\frac{\partial\phi}{\partial t}+\phi\frac{\partial\phi^\star}{\partial t}\right)\]

\[=\frac{-\hbar^2}{2m}\nabla(\phi^\star\nabla\phi-\phi\nabla\phi^\star)-i\hbar\frac{\partial\phi^\star\phi}{\partial t},\]

or, substituting

\[{\bf\mathrm{j}}=\frac{-i\hbar^2}{2m}(\phi^\star\nabla\phi-\phi\nabla\phi^\star),\]

\[\rho=\hbar\phi^\star\phi,\]

we get

\[-i\left(\nabla{\bf\mathrm{j}}+\frac{\partial\rho}{\partial t}\right)=0,\]

\[\nabla{\bf\mathrm{j}}+\frac{\partial\rho}{\partial t}=0.\]

### References

Feynman, Richard P.,The Feynman Lectures on Physics III., Addison-Wesley, 1977

Aitchison, I. J. R. & Hey, A. J. G.,Gauge Theories in Particle Physics, Institute of Physics Publishing, 1996