Advanced Quantum Mechanics

We are given a Hamiltonian: \[ H=\int dx\left[\Psi^\dagger(x)\left(-\frac{\nabla^2}{2m}\right)\Psi(x)+V(x)\Psi^\dagger(x)\Psi(x)\right]. \] Here we take \(\hbar=1\). The first term is the kinetic energy term, because in single-particle quantum mechanics, \(\hat p^2/(2m)\) in the position representation corresponds to \(-\nabla^2/(2m)\). The second term is the potential energy term, where \(\Psi^\dagger(x)\Psi(x)\) is the particle number density at position \(x\), so \(V(x)\Psi^\dagger(x)\Psi(x)\) represents the potential energy density at \(x\). If we take \(V(x)=mc^2\), it just adds a rest energy to each particle; there is still no interaction force between particles here. ...

We start with a setup: the potential energy \(V(x)\) in the Hamiltonian has a shape similar to a symmetric double well, with two symmetric minima and a very high barrier in the middle (insert a diagram of the potential energy here). Let’s discuss the energy eigenstates of this system and the time evolution of the states. First, we need to solve the time-independent Schrödinger equation \(H|\psi\rangle = E|\psi\rangle\) to find the energy eigenvalues and eigenstates. Here, the projection of the eigenstate in the \(x\) basis is the wave function \(\psi(x)=\langle x|\psi\rangle\). ...

Let’s talk about the basic setup of quantum field theory. Start with a single quantum harmonic oscillator. Its energy eigenvalue equation is \[ H|\psi\rangle = E|\psi\rangle . \] It has a set of discrete energy eigenstates. For the harmonic oscillator, a more convenient set of eigenstates is the number states \(|n\rangle\) (\(n=0,1,2,\dots\)), satisfying \(H|n\rangle=E_n|n\rangle\). We introduce the annihilation operator \(a\) and the creation operator \(a^\dagger\), require them to satisfy \([a,a^\dagger]=1\), and define the number operator \(N=a^\dagger a\). The number states are also eigenstates of \(N\): ...

We want to classify different particles using wavefunctions. Consider a two-particle system: if we exchange the two particles, how does the state of the system change? In classical mechanics, even if two particles are completely identical in their physical properties, in principle we can still distinguish them by “labels” (such as “particle 1” and “particle 2”). Therefore, exchanging the positions of two classical particles produces a new, distinguishable microscopic configuration. ...

For the one-dimensional harmonic oscillator problem, its Hamiltonian can be written as \(H = \frac{p^2}{2} + \frac{1}{2}\omega^2 x^2\). Our goal is to solve its energy eigenvalue equation \(H\varphi = E\varphi\). To solve it using a more concise algebraic method, we introduce a pair of ladder operators defined as: \[ a^{\pm} = \frac{p \pm i\omega x}{\sqrt{2\omega}} \] Here \(a^+\) is usually called the creation operator \(a^\dagger\), and \(a^-\) is called the annihilation operator \(a\). With these operators, we can express the Hamiltonian in a more compact form. We define the number operator as \(N = a^\dagger a\). After some derivation, we find that the relation between the Hamiltonian and the number operator is: ...

To solve for the state of a particle in a system with a central potential, such as an electron in an atom, we describe its state using a wavefunction \( \psi(r, \theta, \phi) \) in spherical coordinates. A key mathematical technique here is the separation of variables. This method is applicable because the system’s potential energy is spherically symmetric, meaning it only depends on the distance \(r\) from the center, not on the angles \( \theta \) or \( \phi \). This symmetry allows us to decompose the wavefunction into a product of a radial part \( R(r) \) and an angular part \( Y(\theta, \phi) \). By expressing \( \psi(r, \theta, \phi) = R(r)Y(\theta, \phi) \), we can transform the single complex Schrödinger equation into a set of simpler, one-dimensional ordinary differential equations, which can be solved separately. ...

A symmetry can be understood as an operator that, when applied to a system, leaves its fundamental characteristics unchanged. For instance, a crystal lattice exhibits translational symmetry; shifting its position by a lattice vector does not alter its structure. In quantum mechanics, this concept is deeply tied to the degeneracy of energy levels. Degeneracy means that different quantum states can share the same energy. While symmetry sometimes implies degeneracy, but degeneracy in energy levels is always a sign of an underlying symmetry in the system. ...

In quantum mechanics, the state of a physical system is described by a state vector, denoted using Dirac’s bra-ket notation as \(|\psi\rangle\). These state vectors are elements of a complex vector space called a Hilbert space. Given two state vectors \(|\psi\rangle\) and \(|\phi\rangle\), their inner product is a complex number denoted by \(\langle\phi|\psi\rangle\). If the inner product of two different states is zero, i.e., \(\langle\phi|\psi\rangle = 0\), the states are said to be orthogonal, representing completely independent physical situations. ...