1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
| # Adiabatic Theorem
The **adiabatic theorem** is a foundational concept in [quantum mechanics](/Quantum_mechanics), first articulated by [Max Born](/Max_Born) and [Vladimir Fock](/Vladimir_Fock) in 1928. The theorem posits that a physical system will remain in its instantaneous [eigenstate](/Eigenstate) if subjected to a sufficiently slow perturbation, provided there is a gap between the eigenvalue of the state and the rest of the [Hamiltonian](/Hamiltonian_(quantum_mechanics))'s [spectrum](/Spectrum_of_an_operator). This principle is crucial for understanding how quantum systems evolve under gradually changing conditions, contrasting sharply with rapid changes where the system's spatial probability density remains static due to insufficient time for adaptation.
## Historical Context and Adiabatic Pendulum
At the 1911 Solvay Conference, [Albert Einstein](/Albert_Einstein) presented a lecture on the quantum hypothesis, which posits that the energy \( E \) of atomic oscillators is quantized as \( E = n h \nu \), where \( n \) is an integer, \( h \) is Planck's constant, and \( \nu \) is the frequency. Following Einstein's lecture, [Hendrik Lorentz](/Hendrik_Lorentz) raised a classical objection: if a simple pendulum is shortened by sliding the wire between two fingers, the energy appears to change smoothly, seemingly invalidating the quantum hypothesis for macroscopic systems. Einstein countered that while both the energy \( E \) and frequency \( \nu \) change, their ratio \( \frac{E}{\nu} \) remains conserved, thus preserving the quantum hypothesis. This exchange underscored the importance of adiabatic processes in quantum mechanics, a concept further explored by [Paul Ehrenfest](/Paul_Ehrenfest) in his paper on the adiabatic hypothesis, which Einstein had reviewed prior to the conference.
## Diabatic vs. Adiabatic Processes
### Comparison
| **Diabatic** | **Adiabatic** |
|--------------|---------------|
| Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial probability density remains unchanged. Typically, there is no eigenstate of the final Hamiltonian with the same functional form as the initial state. The system ends in a linear combination of states that sum to reproduce the initial probability density. | Gradually changing conditions allow the system to adapt its configuration, hence the probability density is modified by the process. If the system starts in an eigenstate of the initial Hamiltonian, it will end in the corresponding eigenstate of the final Hamiltonian. |
### Mathematical Formulation
Consider a quantum-mechanical system with an initial Hamiltonian \( \hat{H}(t_0) \) at time \( t_0 \), where the system is in an eigenstate \( \psi(x, t_0) \). As conditions change, the Hamiltonian evolves continuously to \( \hat{H}(t_1) \) at a later time \( t_1 \). The system's evolution is governed by the time-dependent [Schrödinger equation](/Schr%C3%B6dinger_equation), leading to a final state \( \psi(x, t_1) \). The adiabatic theorem states that the nature of this evolution depends critically on the time \( \tau = t_1 - t_0 \) over which the modification occurs.
For a truly adiabatic process, where \( \tau \to \infty \), the final state \( \psi(x, t_1) \) will be an eigenstate of the final Hamiltonian \( \hat{H}(t_1) \), albeit with a modified configuration:
\[ |\psi(x, t_1)|^2 \neq |\psi(x, t_0)|^2. \]
Conversely, in the limit \( \tau \to 0 \), the system undergoes a diabatic process where the configuration remains unchanged:
\[ |\psi(x, t_1)|^2 = |\psi(x, t_0)|^2. \]
The "gap condition" in Born and Fock's original definition refers to the requirement that the spectrum of \( \hat{H} \) is discrete and nondegenerate, ensuring unambiguous ordering of states. In 1999, J. E. Avron and A. Elgart reformulated the adiabatic theorem to accommodate situations without a gap.
## Comparison with Thermodynamics
In [thermodynamics](/Thermodynamics), the term "adiabatic" describes processes without heat exchange between the system and its environment, typically occurring faster than the timescale of heat exchange. In contrast, the classical and quantum mechanics definition aligns more closely with the thermodynamical concept of a [quasistatic process](/Quasistatic_process), where the system remains nearly always at equilibrium. In quantum mechanics, adiabatic processes imply that the timescale of electron and photon interactions is much faster than their propagation timescale, allowing the system to adapt its state and quantum numbers to avoid quantum jumps.
## Example Systems
### Simple Pendulum
Consider a [pendulum](/Pendulum) oscillating in a vertical plane. If the support is moved slowly, the mode of oscillation adapts to the changing conditions, retaining its initial character. This classical example is detailed in the [Adiabatic invariant](/Adiabatic_invariant) page.
### Quantum Harmonic Oscillator
For a more quantum-mechanical example, consider a [quantum harmonic oscillator](/Quantum_harmonic_oscillator) as the spring constant \( k \) is increased. If \( k \) is increased adiabatically (\( \frac{dk}{dt} \to 0 \)), the system remains in an instantaneous eigenstate of the current Hamiltonian, adapting its functional form to the slowly varying conditions. Conversely, a rapid increase in \( k \) results in a diabatic process where the system has no time to adapt, leading to a linear superposition of eigenstates of the new Hamiltonian.
### Avoided Curve Crossing
In a two-level atom subjected to an external magnetic field, the states can be thought of as atomic angular-momentum states. The system's wavefunction can be represented as a linear combination of these diabatic states. As the magnetic field changes, the energies of these states vary, leading to an avoided crossing where the eigenvalues of the Hamiltonian cannot be degenerate. An adiabatic increase in the magnetic field ensures the system remains in an eigenstate of the Hamiltonian, while a diabatic increase leads to a transition between states.
## Mathematical Statement and Proofs
### Mathematical Statement
Under a slowly changing Hamiltonian \( H(t) \) with instantaneous eigenstates \( |n(t)\rangle \) and corresponding energies \( E_n(t) \), a quantum system evolves from the initial state \( |\psi(0)\rangle = \sum_n c_n(0)|n(0)\rangle \) to the final state \( |\psi(t)\rangle = \sum_n c_n(t)|n(t)\rangle \), where the coefficients undergo a phase change:
\[ c_n(t) = c_n(0) e^{i\theta_n(t)} e^{i\gamma_n(t)}, \]
with the dynamical phase \( \theta_m(t) = -\frac{1}{\hbar} \int_0^t E_m(t') dt' \) and the geometric phase \( \gamma_m(t) = i \int_0^t \langle m(t') | \dot{m}(t') \rangle dt' \).
### Proofs
#### Sakurai's Proof
Sakurai's proof in *Modern Quantum Mechanics* begins with the time-independent Schrödinger equation for the instantaneous eigenstates and energies. By expanding the state vector in terms of these eigenstates and applying the time-dependent Schrödinger equation, one derives a coupled first-order differential equation for the coefficients. Neglecting the right-hand side under the adiabatic approximation leads to the adiabatic theorem, showing that the system remains in its initial eigenstate with a phase change.
#### Adiabatic Approximation
The adiabatic approximation is valid when the rate of change of the Hamiltonian is small and there is a finite gap between the energies. This leads to the system evolving independently in its eigenstates, with the geometric phase becoming a gauge-invariant physical quantity known as the [Berry phase](/Berry_phase) for cyclic adiabatic evolutions.
## Example Applications
The adiabatic theorem finds applications in various fields, including:
- **Thermodynamics**: Explaining the temperature dependence of specific heat, thermal expansion, and melting.
- **Transport Phenomena**: Understanding the temperature dependence of electric resistivity in conductors and conductivity in insulators, as well as properties of low-temperature superconductivity.
- **Optics**: Describing optic absorption in the infrared for ionic crystals, Brillouin scattering, and Raman scattering.
## Deriving Conditions for Diabatic vs. Adiabatic Passage
### Diabatic Passage
In the limit \( \tau \to 0 \), the system undergoes a diabatic passage where the functional form remains unchanged:
\[ |\langle x | \psi(t_1) \rangle|^2 = |\langle x | \psi(t_0) \rangle|^2. \]
The validity of this approximation is characterized by the probability \( P_D = 1 - \zeta \), where \( \zeta \) is the probability of finding the system in a different state.
### Adiabatic Passage
In the limit \( \tau \to \infty \), the system evolves adiabatically, adapting its form to the changing conditions:
\[ |\langle x | \psi(t_1) \rangle|^2 \neq |\langle x | \psi(t_0) \rangle|^2. \]
The validity of the adiabatic approximation is given by the probability \( P_A = \zeta \).
## Calculating Adiabatic Passage Probabilities
### The LandauâZener Formula
For a linearly changing perturbation, the LandauâZener formula provides an analytic solution for the diabatic transition probability:
\[ P_D = e^{-2\pi \Gamma}, \]
where \( \Gamma = \frac{a^2/\hbar}{|dq/dt \partial (E_2 - E_1)/\partial q|} \).
### Numerical Approach
For nonlinear changes or time-dependent coupling, numerical solutions to the time-dependent Schrödinger equation are necessary to determine the diabatic transition probability.
## See Also
- [LandauâZener formula](/Landau%E2%80%93Zener_formula)
- [Berry phase](/Berry_phase)
- [Quantum stirring, ratchets, and pumping](/Quantum_stirring,_ratchets,_and_pumping)
- [Adiabatic quantum motor](/Adiabatic_quantum_motor)
- [BornâOppenheimer approximation](/Born%E2%80%93Oppenheimer_approximation)
- [Eigenstate thermalization hypothesis](/Eigenstate_thermalization_hypothesis)
- [Adiabatic process](/Adiabatic_process)
|