List of quantum-mechanical systems with analytical solutions

Quantum mechanics is a part of science that helps us learn about tiny particles like atoms and electrons. One important idea in quantum mechanics is the Schrödinger equation. This equation tells us how particles change over time.

For some special systems, scientists can solve this equation with math instead of using computers. These special systems are called "analytical solutions." They help scientists understand how particles behave.

The list below shows some of these special systems. They give us useful information about the world of quantum physics.

The Schrödinger equation can be made simpler for systems that do not change over time. This is called the time-independent Schrödinger equation. It helps scientists find special states where the system's energy and other properties stay the same. These solutions are 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 science that helps us understand how very small particles behave. Some special systems can be solved exactly with math. This means scientists can predict exactly how they will act.

These systems include simple models like the two-state quantum system and the free particle.

Other systems involve particles in different places, such as boxes, rings, and special movements. Examples include the particle in a box, the quantum harmonic oscillator, and the hydrogen atom. These models help scientists learn about real particles and how they move.

Solutions

The study of quantum mechanics looks at how very small particles behave. One key equation is the Schrödinger equation. It helps us understand the wave-like properties of these particles.

When we search for special, steady states of a system, we use the time-independent Schrödinger equation. This equation helps scientists find the energy levels 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 }