Quantum mechanics/The essential ideas

Quantum mechanics is formally described as a set of abstract postulates from which it is possible to predict what the result is of any measurement on a given system. We will discuss only a simplified set of postulates, followed by a very standard set of examples. The formal postulates with some additional complications are given in Lessons 8-9.

Description of a classical system of particles
In classical physics, a system of N particles is described by a set of N positions and N velocities as a function of time, i.e. {r1(t), r2(t), ..., rn(t), v1(t), v2(t), ... , vn(t)}.

For example, r1(t) is the position of particle 1 as a function of time and v1(t) is its velocity. In 3 dimensions r and v can be vectors with three components along three reference Cartesian axes, i.e. r1(t) = [x1(t), y1(t), z1(t)]. The time evolution of the system is described by Newton's law $$F_{x_1}=m_1 a_{x_1}=m_1 \frac{dv_{x_1}}{dt}$$ (valid for all coordinates). In a conservative system (i.e. one with no friction) the forces can be expressed as derivatives of a function known as the potential energy of the system V. This function depends only on the position of the particles. The relation between potential energy and force is $$F_{x_1}=-\frac{\delta V}{\delta x_1}$$ and should be remembered.

In summary, to describe a classical system of N particles you need: From the potential energy you can calculate the forces and accelerations. From these you can study the evolution of the position and velocities as time goes on.
 * the potential energy
 * the initial position and velocities.

An important characteristic of conservative systems is that the sum of kinetic and potential energy is conserved at all times, and this is the total energy of the system. In one dimension: $$E_{kinetic} + E_{potential} = \frac{1}{2} m\left( \frac{dx}{dt} \right)^2 + V\left(x\right) = constant$$ (Eq. 1)

Description of a quantum system of particles
A system at time t is described by a wavefunction ψ(r, t) and not by a set of positions and velocities.

ψ(r, t) is a function of many variables: t is time, and r without subscripts represents all possible position coordinates. So for a system containing only one particle in one dimension, you have ψ(x, t); for a system containing one particle in 3 dimensions, you have ψ(x, y, z, t); for a system containing two particles in 3 dimensions, you have ψ(x1, y1, z1, x2, y2, z2, t) and so on. The values of ψ(r, t) can be complex: there is only one value of ψ(r, t) for each of the variables (ψ is a single valued function); ψ(r, t) must be continuous and variable.

The time evolution of the wavefunction is given by the Schrödinger equation.

For a one dimensional system the Schrödinger equation takes the form $$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \left \lbrace - \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right \rbrace \Psi (x,t)$$

For the moment, it is sufficient to notice that the potential energy V(x) appears in the equation.

The probability of finding a particle in one particular position x at a given time is proportional to |Ψ(x, t)|2.

More definitively, the probability of finding a particle between x and x+dx is given by: $$\frac{|\Psi (x,t)|^2 dx}{\int_{-\infty}^{+\infty} |\Psi (x,t)|^2\, dx}$$ (Eq. 2)

Quantum mechanics does not predict the result of a measurement but rather the probability of it. This is true for measuring the position as well as all other measurable quantities (named observables: energy, momentum, etc.) An initial comparison between classical and quantum mechanics is given in the table below.

General structure in 1D
The Schrödinger equation above can be written in more compact form as $$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \hat{H} \Psi (x,t)$$  (Eq. 3)

where Ĥ is called the Hamiltonian operator: $$\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$$ (Eq. 4)

The first term of the sum is called the kinetic energy operator. It is somewhat analogous to equation 1, where the total energy is expressed as the sum of the kinetic energy and the potential energy. The mathematical object Ĥ must have something to do with the total energy. Ĥ contains all the information needed to describe a given system.

General solution and time independent version
We can try to simplify the solution of equation 3 by assuming that the solution can be written as Ψ(x, t)=ψ(x)g(t), i.e. as the product of a function of only the position coordinate and a function of only the time coordinate. Substituting this into equation 3 we get: $$i\hbar \frac{\partial \psi(x)g(t)}{\partial t} = \hat{H}(x) \psi(x) g(t)$$ (Eq. 5)

where Ĥ(x) is used instead of Ĥ to stress that the Hamiltonian does not depend on time. It is easy to see now that equation 5 $$i\hbar\psi(x) \frac{\partial g(t)}{\partial t} = g(t) \left \lbrack \hat{H}(x) \psi(x) \right \rbrack$$ (Eq. 6)

and you can therefore separate the terms containing x and those containing t: $$i \hbar \frac{ \frac{\partial g(t)}{\partial t}}{g(t)} = \frac{\left \lbrack \hat{H}(x) \psi(x) \right \rbrack}{\psi (x)}$$ (Eq. 7)

The left side of equation 7 is a function of t only, while the right hand side is a function of x only. The only way they can be identical is if they are both equal to a constant which we indicate as E.

$$i\hbar \frac{ \frac{\partial g(t)}{\partial t}}{g(t)} \to i\hbar \frac{\partial g(t)}{\partial t} = E g(t)$$ (Eq. 8)  TIME PART

$$\frac{ \left \lbrack \hat{H} (x) \psi (x) \right \rbrack }{\psi (x)} \to \hat{H} (x) \psi (x) = E \psi (x)$$ (Eq. 9)  SPACE PART

The solution of the time part (equation 8) is very simple (check it!): g(t) = exp(-iEt / ħ) (Eq. 10).

The solution of the space part (equation 9) will be the main subject of the rest of the Lessons: Ĥ(x) = Eψ(x) (Eq. 11). This is called the time independent Schrödinger equation. It is known as an eigenvalue equation and it has solution only for a set of discrete values of E, i.e. E1, E2, E3,... which are called the eigenvalues of the equation. For each eigenvalue, e.g. E1, there is a corresponding eigenfunction, e.g. ψ1, which satisfies the equation. The solution to equation 3 can be written as: $$\begin{matrix}\hat{H} (x) \psi_1 (x) & = & E_1 \psi_1 (x) \\ \hat{H} (x) \psi_2 (x) & = & E_2 \psi_2 (x) \\ \ & \cdots \ \\ \hat{H} (x) \psi_j (x) & = & E_j \psi_j (x) \end{matrix}$$ (Eq. 12)

Remembering that the general solution of the Schrödinger equation was Ψ(x, t)=ψ(x)g(t), you can write the time dependent solutions as: $$\begin{matrix}\Psi_1 (x,t) & = & \psi_1 (x) \exp (\frac{-i E_1 t}{\hbar}) \\ \Psi_2 (x,t) & = & \psi_2 (x) \exp (\frac{-i E_2 t}{\hbar}) \\ \ & \cdots \ \\ \Psi_j (x,t) & = & \psi_j (x) \exp (\frac{-i E_j t}{\hbar}) \end{matrix}$$ (Eq. 13)

Considering that the time dependent part of the wavefunction is always the same, any system can be fully described if we find the solution to the time independent Schrödinger equation.

We can identify the eigenvalues E1, E2, E3,... with the energy of the system described by the wavefunction Ψ1, Ψ2, Ψ3,...

Pure and mixed states: an additional complication
Not only Ψ1, Ψ2, Ψ3,... of equation 13 are solutions of equation 3, but also any linear combination of them is a solution, i.e. any expression such as $$\Psi (x,t) = C_1 \Psi_1 (x,t) + C_2 \Psi_2 (x,t) + C_3 \Psi_3 (x,t) + \cdots$$ (Eq. 14)

$$\Psi (x,t) = C_1 \psi_1 (x) \exp (\frac{-i E_1 t}{\hbar}) + C_2 \psi_2 (x) \exp (\frac{-i E_2 t}{\hbar}) + C_3 \psi_3 (x) \exp (\frac{-i E_3 t}{\hbar}) + \cdots$$

where the coefficients C1, C2, C3 can take any complex value.

When a time dependent wavefunction contains only one eigenfunction of the Hamiltonian, i.e. all the coefficients are zero except one, the system is said to be in a pure state. Otherwise, if there is more than one eigenfunction contributing to the wavefunction, the system is in a mixed state.

If we measure the energy of a mixed state the result is NOT some average energy of the contributing states. Such a measurement will always give as a result one of the eigenvalues of the Hamiltonian E1, E2, E3,... The probability that the result of the measurement is E1, E2, E3,... is proportional to |C1|², |C2|², |C3|²,..., the modulus squared of the coefficients appearing in equation 14.

Summary
The discussion of pure and mixed states above is one of the aspects of quantum mechanics that is more difficult to understand and will be reconsidered in detail towards the end of the course. For the moment, it is sufficient to remember the following point: We will leave aside for now the problem of how a system ends up in a pure or a mixed state and we will describe only pure states for the next Lessons.
 * Solving the time independent Schrödinger equation Ĥψ = Eψ is the most important and difficult step to describe a quantum mechanical system.
 * The solution of the time independent Schrödinger equation is a discrete set of eigenvalues E1, E2, E3,... and corresponding eigenfunctions.
 * If we measure the energy of a system we will obtain as a result one of the eigenvalues of the Hamiltonian, i.e. the system appears to have discrete energy levels.


 * 1) The kinetic energy operator for a single particle is $$- \frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}$$ in one dimension, $$- \frac{\hbar^2}{2m} \left ( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right )$$ in two dimensions, and $$- \frac{\hbar^2}{2m} \left ( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right )$$ in three dimensions. If there are two particles, you have to write the sum of the kinetic energy of each particle. For example, the total kinetic energy of a particle of mass m1 and a particle of mass m2 in one dimension is $$- \frac{\hbar^2}{2m_1} \frac{\partial^2}{\partial x_1^2} - \frac{\hbar^2}{2m_2} \frac{\partial^2}{\partial x_2^2}$$. Write the kinetic energy for a system in 3D containing an electron and a proton.
 * 2) Partial derivatives are fairly simple to calculate. Given $$f(x,y) = x^2 + xy^2 + y^3 + 2$$, calculate $$\frac{\partial}{\partial x} f(x,y)$$ and $$\frac{\partial}{\partial y} f(x,y)$$. Now calculate the second partial derivatives $$\frac{\partial^2}{\partial x^2} f(x,y)$$ and $$\frac{\partial^2}{\partial y^2} f(x,y)$$.
 * 3) Repeat the proof that the wavefunction can be divided into a time dependent and a time independent part in the case of a Hamiltonian function of two spatial coordinates Ĥ(x, y).

Next: Lesson 3 - Wavepackets