Quantum Mechanics is a theory that describes how tiny stuff like electrons move around. It’s like Newton’s Law: F=ma, which only works for the macroscopic objects, but is still valid in the microscopic world.

Why is it called “Quantum” Mechanics?

If you are learning Quantum Mechanics for the ﬁrst time, or you are asked by a friend who wants to know about Quantum Mechanics, the ﬁrst question you might have probably is why it is called “Quantum” Mechanics? A quick answer to this question is that Quantum Mechanics is a theory that can explain various quantized phenomena we encountered in experiments, which cannot be explained by Classical Mechanics. For example, the allowed energy level of the electron in the Hydrogen atom is quantized, i.e., we have 1s, 2s, 2p, …… all those discrete allowed orbitals with quantized or discrete allowed energies.

Although “Quantum” is one of the possible outcomes of Quantum Mechanics, quantization is not built inside the Quantum Mechanics. The “quantized” property comes from the fact that you are asking the eigenvalue problem of a bounded system, or you can say it comes from the boundary condition. For example, the allowed total energy of the electron in a Hydrogen atom is quantized is due to the fact that the electron is bounded by the Coulomb potential from the proton. Also, the quantization of a component of Spin (e.g., the eigenvalue of S_z) is due to the periodic boundary condition of spacial rotation (i.e., exp{i2π}=exp{i0} or the world should look similar after a 360° rotation). For free particles in Quantum Mechanics, there is nothing really quantized. The momentum and the energy of a free particle are continuous.

Being quantized as a result of the boundary condition is not that mysterious and is not exclusive to Quantum Mechanics. In classical physics, we also have a very similar situation. For example, the vibration modes of a classical string with two fixed ends are quantized (e.g., the string in a violin). It makes the sound waves which are the superposition of its base frequency and all harmonics that are frequencies equal to the integer multiples of the base frequency.

Of course, here I’m talking about the original meaning of the word “Quantum.” Nowadays, people just use the word “quantization” as a technical term referring to “Quantum-Mechanicalization.” For instance, the jargon Canonical Quantization in Quantum Mechanics means that we take the commutator [x, p] ≡ xp–px = iℏ, which was equal to zero in the classical case. In other words, it’s a process that makes the order of performing position and momentum operators matter, thereby making the system “Quantum-Mechanicalized.”

By the way, you might guess that the existence of photon, which is the quantized electromagnetic waves, is also a result of Quantum Mechanics. However, problems involving photons are not usually addressed in the standard Quantum Mechanics. In fact, to fully understand what photon is, we need a theory called Quantum Field Theory (QFT), which is the quantum version of classical field theory. (Especially, the Quantum Electrodynamics, the QED, which is the quantum version of Electrodynamics.) In QED, photons are treated as the quantized electromagnetic waves, while electrons are treated as the quantized Dirac field of electron. This also explains why electrons are identical particles because all electrons are the excitation of a single quantized Dirac field.

Probabilistic

On the other hand, there is another well-known feature about things that happen in the quantum world. That is, everything is probabilistic.

In Classical Mechanics, considering a 1-dimensional case, we can describe the state of a particle by its position x and its velocity v as (x, v). For a free particle with mass m moving with a constant velocity v, its initial state is (x₀, v), and it is evolved with time t as (x₀ + vt, v). If you try to measure the position, momentum, and kinetic energy of such particle simultaneously at time t₁, you will deﬁnitely get x₀ + v t₁, mv, and mv²/2 with a hundred percent conﬁdence with no doubt. Also, the state of the particle remains (x₀ + vt, v) after the measurement. The particle keeps moving along its original path. The measurement does not affect the state of the particle. However, this is not the case in Quantum Mechanics.

For the electron in a Hydrogen atom that is orbiting around the proton in the orbital 1s, we can say the state of the electron is | 1s 〉. The state | 1s 〉 completely describes how the electron is moving around the space just like the state (x₀ + vt, v) we encountered for the classical particle (Although neither of them described the rotational property of the electron or the classical particle we mentioned). If you try to measure the total energy of the electron, that is ﬁne. You will always get E₁s = −13.6eV. Also, after the measurement, the electron stays in the state | 1s 〉. It seems nothing looks surprising, but that is because the state | 1s 〉 is an eigenstate of the total energy measurement or the Hamiltonian operator.

Nevertheless, if you try to measure the position of this electron, something mysterious happens. First, although you will get a position of the electron, say x₁, you can’t predict which position you will get before you actually measure it. The only thing you are able to predict is the probability of ﬁnding this electron at position x₁, which can be written as P₁s(x₁)=|〈 x₁|1s 〉|². The notation 〈 x₁|1s 〉 means the inner product of two state vectors | x₁ 〉and| 1s 〉and if you are interested about what is that mean exactly, you’ll need to read a Quantum Mechanics textbook. The fact that you can only predict the probability of ﬁnding the 1s electron in a certain position x₁ is equivalent to that there is no “path” or “trajectory” of a quantum particle. This is also why you always see the ﬁgure of 1s orbital on the textbook as some “electron cloud” around the proton instead of a clear trajectory following an elliptic orbit like how the Earth is orbiting around the Sun. Second, after the measurement, the state of this electron collapses or jumps into the state | x₁ 〉. So, it’s no longer orbiting around the proton! Third, if you try to measure the position of the electron again right after the previous measurement, you will still get x₁ with a hundred percent probability. This is because the electron is now in the state | x₁ 〉, which is an eigenstate of the position measurement. Fourth, since this electron is no longer in the state | 1s 〉 but the state | x₁ 〉, if you try to measure its total energy again at this point, you won’t (always) get −13.6 eV. That is because the state | x₁ 〉is not a total energy eigenstate. In fact, the probability you get the value −13.6eV is |〈 1s|x₁ 〉|²

The fact that the results of measurements are probabilistic and the exact answer of a given measurement is only known after you really did the measurement is one of the biggest mysteries of Quantum Mechanics. Because prior to Quantum Mechanics, we always think the quantities we are going to measure are the physical reality of the system. They should be already there no matter you are going to measure it or not. This probabilistic nature of Quantum Mechanics also bothers Einstein a lot, such that he actually wrote the famous paper — Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? — with Podolsky and Rosen (a.k.a. the EPR paper). This also leads to the famous quote, “Is the moon there when nobody looks?”

Wave Mechanics vs. Matrix Mechanics

You probably heard that there are two different formulations of Quantum Mechanics. One is the Wave Mechanics, which was mainly developed by Erwin Schrödinger in 1926. The other is the Matrix Mechanics, which was mainly developed by Werner Heisenberg in 1925. It’s hard to believe at the ﬁrst time that how the “matrix” can be used for doing Mechanics. However, they are actually equivalent. The wavefunction ψ(x), which is a function, can be represented by a vector. This can be done by identifying the value of ψ(x) as the “x-th” “component” of the corresponding vector. Since the variable x in ψ(x) is continuous and running from -∞ to +∞, there are infinite numbers of components in such vector. So, the vector we are dealing with here is essentially infinite-dimensional. Furthermore, the differential operators we encounter in Schrodinger’s wave equation can be represented by Matrices, which is also infinite-dimensional. So, differentiation becomes the matrix product, and solving linear differential equations in the Wave Mechanics becomes solving linear algebra problems in the Matrix Mechanics!