List of quantum-mechanical systems with analytical solutions

Quantum mechanics is a fascinating area of science that helps us understand how tiny particles, like atoms and electrons, behave. One of the key equations in quantum mechanics is the Schrödinger equation, which describes how these particles change over time. Usually, solving this equation requires complex computer calculations, but for some special systems, scientists can find exact solutions using math.

These special systems have something called "analytical solutions," meaning their behavior can be described with precise mathematical formulas. Finding these solutions helps scientists learn a lot about how particles work and why they behave the way they do. The list below shows some of these special systems, giving us valuable insight into the world of quantum physics.

The Schrödinger equation can be simplified for systems that do not change over time, leading to what is called the time-independent Schrödinger equation. This equation helps scientists find specific states, known as stationary states, where the system's energy and other properties remain constant. These solutions are very important because they show us exactly how particles are arranged and how much energy they have in these stable states.

Solvable systems

Quantum mechanics is a fascinating area of science that helps us understand how tiny particles behave. Some special systems can be solved exactly using mathematics, which means scientists can predict their behavior perfectly. These systems include simple models like the two-state quantum system and the free particle.

Other solvable systems involve particles in different kinds of potentials, such as boxes, rings, and harmonic oscillators. For example, there is the particle in a box, the quantum harmonic oscillator, and the hydrogen atom. These models help scientists learn about real particles and their movements in various situations.

Solutions

The study of quantum mechanics often involves solving equations that describe how tiny particles behave. One important equation is the Schrödinger equation, which helps us understand the wave-like properties of these particles.

When we look for special, unchanging states of a system, we use the time-independent Schrödinger equation. This equation helps scientists find possible energy levels that a particle can have, showing how particles can exist in different states.

System	Hamiltonian	Energy	Remarks
Two-state quantum system	α I + r σ ^ {\displaystyle \alpha I+\mathbf {r} {\hat {\mathbf {\sigma } }}\,}	α ± \| r \| {\displaystyle \alpha \pm \|\mathbf {r} \|\,}
Free particle	− ℏ 2 ∇ 2 2 m {\displaystyle -{\frac {\hbar ^{2}\nabla ^{2}}{2m}}\,}	ℏ 2 k 2 2 m , k ∈ R d {\displaystyle {\frac {\hbar ^{2}\mathbf {k} ^{2}}{2m}},\,\,\mathbf {k} \in \mathbb {R} ^{d}}	Massive quantum free particle
Delta potential	− ℏ 2 2 m d 2 d x 2 + λ δ ( x ) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+\lambda \delta (x)}	− m λ 2 2 ℏ 2 {\displaystyle -{\frac {m\lambda ^{2}}{2\hbar ^{2}}}}	Bound state
Symmetric double-well Dirac delta potential	− ℏ 2 2 m d 2 d x 2 + λ ( δ ( x − R 2 ) + δ ( x + R 2 ) ) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+\lambda \left(\delta \left(x-{\frac {R}{2}}\right)+\delta \left(x+{\frac {R}{2}}\right)\right)}	− 1 2 R 2 ( λ R + W ( ± λ R e − λ R ) ) 2 {\displaystyle -{\frac {1}{2R^{2}}}\left(\lambda R+W\left(\pm \lambda R\,e^{-\lambda R}\right)\right)^{2}}	ℏ = m = 1 {\displaystyle \hbar =m=1} , W is Lambert W function, for non-symmetric potential see here
Particle in a box	− ℏ 2 2 m d 2 d x 2 + V ( x ) {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)} V ( x ) = { 0 , 0	π 2 ℏ 2 n 2 2 m L 2 , n = 1 , 2 , 3 , … {\displaystyle {\frac {\pi ^{2}\hbar ^{2}n^{2}}{2mL^{2}}},\,\,n=1,2,3,\ldots }	for higher dimensions see here
Particle in a ring	− ℏ 2 2 m R 2 d 2 d θ 2 {\displaystyle -{\frac {\hbar ^{2}}{2mR^{2}}}{\frac {d^{2}}{d\theta ^{2}}}\,}	ℏ 2 n 2 2 m R 2 , n = 0 , ± 1 , ± 2 , … {\displaystyle {\frac {\hbar ^{2}n^{2}}{2mR^{2}}},\,\,n=0,\pm 1,\pm 2,\ldots }
Quantum harmonic oscillator	− ℏ 2 2 m d 2 d x 2 + m ω 2 x 2 2 {\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+{\frac {m\omega ^{2}x^{2}}{2}}\,}	ℏ ω ( n + 1 2 ) , n = 0 , 1 , 2 , … {\displaystyle \hbar \omega \left(n+{\frac {1}{2}}\right),\,\,n=0,1,2,\ldots }	for higher dimensions see here
Hydrogen atom	− ℏ 2 2 μ ∇ 2 − e 2 4 π ε 0 r {\displaystyle -{\frac {\hbar ^{2}}{2\mu }}\nabla ^{2}-{\frac {e^{2}}{4\pi \varepsilon _{0}r}}}	− ( μ e 4 32 π 2 ϵ 0 2 ℏ 2 ) 1 n 2 , n = 1 , 2 , 3 , … {\displaystyle -\left({\frac {\mu e^{4}}{32\pi ^{2}\epsilon _{0}^{2}\hbar ^{2}}}\right){\frac {1}{n^{2}}},\,\,n=1,2,3,\ldots }