Imaginary Time Evolution vs Real Time Evolution in Quantum Mechanics

Last Updated on June 15, 2025 by Sushanta Barman

Quantum mechanics describes how quantum systems change over time. There are two main ways to model this evolution: real time evolution and imaginary time evolution. While both use the same mathematical framework (the Schrödinger equation), they serve different purposes in physics and computational simulations.

Real Time Evolution

Real time evolution is what happens physically in nature. It follows the time-dependent Schrödinger equation:

\[i\hbar \frac{\partial \psi(x, t)}{\partial t} = \hat{H} \psi(x, t),\]

where, \(\psi(x, t)\) represents the quantum state (or wave function) at position \(x\) and time \(t\), \(\hat{H}\) denotes the Hamiltonian operator, which corresponds to the total energy of the system, and \(i\) is the imaginary unit.

Physical Meaning

Real time evolution describes the actual, observable dynamics of a quantum system—how a wave function spreads, interferes, or interacts with potentials.

Example: Free Particle Evolution

For a free particle, an initial wave packet spreads out over time, showing dispersion. This process is governed by real time evolution and can be visualized as the probability distribution moving and changing shape.

Imaginary Time Evolution

Imaginary time evolution is a mathematical transformation used mainly as a computational tool to find the ground state (lowest energy state) of a system.

We substitute real time \(t\) with imaginary time \(\tau\) by setting \(\tau = it\):

\[\frac{\partial \psi(x, \tau)}{\partial \tau} = -\frac{1}{\hbar} \hat{H} \psi(x, \tau),\]

Physical Meaning

Imaginary time can be understood as a mathematical tool analogous to cooling the system. Although imaginary time evolution does not correspond to any physical process, it acts as a filter—damping out higher energy components and isolating the ground state after sufficient evolution.

Example: Ground State Finding

Suppose you start with any wave function that has some overlap with the ground state. As you evolve it in imaginary time, all components except the ground state decay away:

\[\psi(x, \tau) = \sum_n c_n \phi_n(x) e^{-E_n \tau/\hbar}\]

As \(\tau\) increases, only the ground state (with lowest \(E_n\)) survives. Moreover, during the imaginary time evolution, normalization is typically performed at each step to prevent the wave function from decaying to zero. This is widely used in computational quantum mechanics (e.g., Quantum Monte Carlo, DFT).

Key Differences

Aspect	Real Time Evolution	Imaginary Time Evolution
Purpose	Describes physical quantum dynamics	Finds ground states numerically
Equation	\(i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi\)	\(\frac{\partial \psi}{\partial \tau} = -\frac{1}{\hbar} \hat{H} \psi\)
Nature	Oscillatory, unitary (probability conserved)	Damped, non-unitary (probability decays without normalization)
Outcome	Evolution of any state over time	Suppression of all states except the ground state
Physical?	Yes	No (mathematical tool)
Typical Use	Quantum dynamics, time evolution, interference	Ground state search, quantum Monte Carlo

Examples

Quantum Harmonic Oscillator

• Real time: If you initialize a superposition of the first and second eigenstates, you will see oscillations between the two states as time progresses.
• Imaginary time: Start with a random function. After evolving in imaginary time and normalizing after each step, the function will converge to the well-known Gaussian ground state. A Python code to find the ground state of a 1D quantum harmonic oscillator is discussed here.

Conclusion

Real time evolution simulates how quantum systems actually behave and change in the physical world. Imaginary time evolution, on the other hand, is a powerful computational shortcut for isolating ground states by mathematically filtering out all higher energy contributions.

Both are indispensable in quantum mechanics, but they serve very different roles.

Further Reading

[1] Sushanta Barman, “Imaginary Time Evolution Method for Finding the Ground State,” MatterWaveX.com, (2024).

[2] C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1 & 2, Wiley-VCH, New York (2005).

[3] D. J. Griffiths, Introduction to Quantum Mechanics, 2nd ed., Pearson Prentice Hall, Upper Saddle River, NJ (2005).