# Exploring Non-Hermitian Hamiltonians for Particle Decay and Quasi-Bound States

• wofsy
In summary, the decay of a particle is seen as an exponential decrease in the amplitude of its state. However, this is an approximation and a more rigorous model would include both the particle and its decay products.
wofsy
Can someone explain how non-Hermitian Hamiltonians are used to account for particle decay?

Hermitian operators always have real eigenvalues. Non-Hermitian operators can have complex eigenvalues. The evolution of an eigenstate of the Hamiltonian is ~exp(-iEt) where E is the energy. If E is complex, there could be a decay.

genneth said:
Hermitian operators always have real eigenvalues. Non-Hermitian operators can have complex eigenvalues. The evolution of an eigenstate of the Hamiltonian is ~exp(-iEt) where E is the energy. If E is complex, there could be a decay.

I infer from you terse reply that the decay is seen in the imaginary part of the eigen-value. It becomes an exponential decay of the state amplitudes. Yes?

wofsy said:
I infer from you terse reply that the decay is seen in the imaginary part of the eigen-value. It becomes an exponential decay of the state amplitudes. Yes?

Yes, imaginary eigenvalues mean that the probability of finding the particle decreases exponentially with time (decay). However, you should keep in mind that this is an approximate way to study decays. In this approach the probability is not conserved and the evolution is non-unitary, which contradicts basic postulates of quantum mechanics.

If you want to have a rigorous model of decays, you'll need to form a bigger Hilbert space, which includes states of both unstable particle and its decay products. In this Hilbert space the decay can be described by a Hermitian Hamiltonian and unitary time evolution operator. The total probability will be conserved, as required. The decreasing probability of finding the particle will be compensated by increasing probability of finding decay products.

meopemuk said:
Yes, imaginary eigenvalues mean that the probability of finding the particle decreases exponentially with time (decay). However, you should keep in mind that this is an approximate way to study decays. In this approach the probability is not conserved and the evolution is non-unitary, which contradicts basic postulates of quantum mechanics.

If you want to have a rigorous model of decays, you'll need to form a bigger Hilbert space, which includes states of both unstable particle and its decay products. In this Hilbert space the decay can be described by a Hermitian Hamiltonian and unitary time evolution operator. The total probability will be conserved, as required. The decreasing probability of finding the particle will be compensated by increasing probability of finding decay products.

very cool. Thanks

You could also consider cases where particles are not bound perfectly, but within a finite barrier. In such case you could find so called meta stable eigen states, which have imaginary components. The Hamiltonian is Hermitian though, but the boundary conditions are wave like at the boundary. Here the decay of the wave function is compensated by the fact that a current is produced outwards (or inwards) at the quasi bound domain, i.e.,

$$\frac{d\mid\Psi\mid^2}{dt}+\nabla\cdot\vec{j}=0$$

where the quantum current j is non vanishing. There are some simple cases where you could solve this exactly, like V=V0 for a<x<a+d and V=infinite at x<0 and else V=0. Eigenstates are normally degenerated symmetrically so that E=E0+/-i*Ej which means yo have both decaying and growing solution in time.

## 1. What is a Non-Hermitian Hamiltonian?

A Non-Hermitian Hamiltonian is a mathematical operator used in quantum mechanics to describe the dynamics of a physical system. It is non-Hermitian because it does not satisfy the Hermitian property, which states that the operator must be equal to its own adjoint. This type of Hamiltonian is often used to study systems with open boundaries or dissipative effects.

## 2. How is a Non-Hermitian Hamiltonian different from a Hermitian Hamiltonian?

A Hermitian Hamiltonian is a special case of a Non-Hermitian Hamiltonian where the operator is equal to its own adjoint. This means that all its eigenvalues are real and its eigenvectors are orthogonal. In contrast, a Non-Hermitian Hamiltonian can have complex eigenvalues and non-orthogonal eigenvectors. This difference leads to distinct physical properties and behaviors of systems described by these two types of Hamiltonians.

## 3. What are some applications of Non-Hermitian Hamiltonians?

Non-Hermitian Hamiltonians have various applications in quantum mechanics, condensed matter physics, and optics. They are commonly used to study open quantum systems, such as those in contact with a dissipative environment. Non-Hermitian Hamiltonians are also used in the study of topological phases of matter, as well as in the design of photonic devices with unique properties.

## 4. How are Non-Hermitian Hamiltonians related to PT symmetry?

PT symmetry is a mathematical symmetry that relates a system to its mirror image when both parity (P) and time (T) are reversed. Non-Hermitian Hamiltonians with PT symmetry have real eigenvalues and can exhibit interesting physical phenomena, such as unidirectional invisibility and gain-induced transparency. However, not all Non-Hermitian Hamiltonians have PT symmetry, and there are ongoing debates about its significance in quantum mechanics.

## 5. Can Non-Hermitian Hamiltonians be experimentally realized?

Yes, Non-Hermitian Hamiltonians have been experimentally realized in various systems, including photonic lattices, coupled resonators, and superconducting circuits. These experiments have provided evidence for the unique properties and behaviors predicted by Non-Hermitian Hamiltonians, such as non-reciprocal effects and exceptional points. The experimental realization of Non-Hermitian Hamiltonians has opened up new avenues for studying and engineering quantum systems with unexpected properties.

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