Coherent-State Preview and the Classical Oscillator Limit
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You should be comfortable with the ladder operator formalism for the harmonic oscillator and the definition of the number states \(|n\rangle\). Familiarity with the expectation values of position and momentum in stationary states is also assumed.
Why this matters
Quantum mechanics must reproduce classical mechanics in the appropriate limit, and coherent states provide the precise bridge between the two.
While the energy eigenstates of the harmonic oscillator are stationary and do not resemble classical motion, specific superpositions of these states—known as coherent states—exhibit oscillatory behaviour that mirrors a classical mass on a spring. This lecture previews the construction of these states and demonstrates how Ehrenfest's theorem guarantees that their expectation values follow Newton's laws of motion, thereby resolving the apparent disconnect between quantum formalism and classical intuition.
The Classical Ideal in Quantum Language
In classical mechanics, a harmonic oscillator is defined by a position \(x(t)\) that oscillates sinusoidally with a fixed amplitude and phase. The energy is continuous, and the particle is always found at a specific point in space. In quantum mechanics, the stationary states \(|n\rangle\) are eigenstates of the Hamiltonian. A key feature of these states is that their probability density \(|\psi_n(x)|^2\) is time-independent. If you measure the position of a particle in an energy eigenstate, the distribution of outcomes does not change with time, regardless of the energy level. This static probability cloud is fundamentally different from the moving particle of classical physics. To recover the classical picture, we must move beyond single energy eigenstates and consider superpositions. The most natural superposition that mimics classical motion is one where the probability density itself oscillates back and forth, maintaining its shape while translating in space. This specific type of superposition is the coherent state. It is the quantum state that most closely resembles a classical oscillator because it minimises the uncertainty product and has a well-defined phase relationship between position and momentum.
Construction via Ladder Operators
Recall from the ladder operator formalism that the position and momentum operators can be expressed in terms of the annihilation operator \(\hat{a}\) and its adjoint \(\hat{a}^\dagger\). Specifically, the position operator is given by \(\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}(\hat{a} + \hat{a}^\dagger)\). A coherent state \(|\alpha\rangle\) is defined as an eigenstate of the annihilation operator \(\hat{a}\), such that \(\hat{a}|\alpha\rangle = \alpha|\alpha\rangle\), where \(\alpha\) is a complex number. This is distinct from the number states \(|n\rangle\), which are eigenstates of the number operator \(\hat{N} = \hat{a}^\dagger\hat{a}\). The parameter \(\alpha\) encodes both the amplitude and the phase of the oscillation. By expanding the coherent state in the number state basis, we find that it is a superposition of all energy eigenstates, with coefficients given by a Poisson distribution. The probability of finding the system in the \(n\)-th energy level is \(P(n) = e^{-|\alpha|^2} \frac{|\alpha|^{2n}}{n!}\). This distribution peaks at \(n \approx |\alpha|^2\), meaning that for large \(|\alpha|\), the state is dominated by high-energy components, which is the regime where classical behaviour emerges.
Time Evolution and Classical Trajectories
The time evolution of a quantum state is governed by the Schrödinger equation. For a harmonic oscillator with Hamiltonian \(\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + \frac{1}{2})\), the time-evolved coherent state is \(|\alpha(t)\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle\). The global phase factor \(e^{-i\omega t/2}\) has no physical effect, but the phase rotation of \(\alpha\) is crucial. The expectation value of the position operator in this state is \(\langle x(t) \rangle = \sqrt{\frac{2\hbar}{m\omega}} \text{Re}(\alpha e^{-i\omega t})\). If we write \(\alpha = A e^{i\phi}\), this becomes \(\langle x(t) \rangle = \sqrt{\frac{2\hbar}{m\omega}} A \cos(\omega t - \phi)\). This is precisely the equation of motion for a classical harmonic oscillator with amplitude \(X_0 = \sqrt{\frac{2\hbar}{m\omega}} A\) and phase \(\phi\). Similarly, the expectation value of the momentum is \(\langle p(t) \rangle = -\sqrt{2m\hbar\omega} A \sin(\omega t - \phi)\). These two equations satisfy the classical relation \(p = m \dot{x}\). Thus, the centre of the quantum probability wave packet moves exactly like a classical particle.
Ehrenfest's Theorem and the Classical Limit
The agreement between the expectation values and classical trajectories is not a coincidence but a consequence of Ehrenfest's theorem. This theorem states that the time derivative of the expectation value of position is equal to the expectation value of momentum divided by mass, and the time derivative of the expectation value of momentum is equal to the negative expectation value of the force. For the harmonic oscillator, the force is linear in position, \(F = -kx\), so the expectation value of the force is simply \(-k\langle x \rangle\). This leads to the differential equation \(\frac{d^2}{dt^2}\langle x \rangle = -\omega^2 \langle x \rangle\), which is identical to the classical equation of motion. The theorem holds for any state, not just coherent states. However, for non-coherent states, the wave packet may spread or distort over time, causing the expectation value to deviate from a simple sinusoidal path or for the uncertainty to grow. Coherent states are special because they do not spread at all; their width remains constant at the ground state width \(\Delta x = \sqrt{\frac{\hbar}{2m\omega}}\). This stability ensures that the wave packet remains localised, allowing the expectation values to track the classical trajectory accurately. The classical limit is therefore achieved when the action \(S\) is much larger than \(\hbar\), which corresponds to large quantum numbers \(n\) or large \(|\alpha|\), where the relative uncertainty \(\Delta x / \langle x \rangle\) becomes small.
The derivations
The Harmonic Oscillator via Ladder Operators
Diagonalises the harmonic oscillator algebraically, deriving the equally spaced spectrum and the zero-point energy from raising and lowering operators.
Ehrenfest's Theorem and the Classical Limit
Derives the equations of motion for expectation values, showing they mirror Newton's laws and pin down the correspondence with classical mechanics.
A common misconception is that the classical limit requires the wave function to collapse to a delta function. In reality, the wave packet of a coherent state retains a finite width \(\Delta x\) even as the amplitude grows. The classical limit is achieved not by the wave packet shrinking to zero width, but by the width becoming negligible compared to the oscillation amplitude. The probability density remains Gaussian, but its centre moves with high precision relative to its size. This distinction is crucial for understanding why quantum effects do not vanish entirely in macroscopic systems, but rather become statistically insignificant relative to the scale of motion. C
Worked example
Problem. Consider a harmonic oscillator with mass \(m = 1.0\ \mathrm{kg}\) and angular frequency \(\omega = 10\ \mathrm{rad/s}\). A coherent state is prepared with parameter \(\alpha = 10\). Calculate the classical amplitude of oscillation \(X_0\) and the uncertainty in position \(\Delta x\). Determine the ratio \(\Delta x / X_0\) to quantify the classicality of the state.
- Calculate the ground state length scale \(x_0 = \sqrt{\frac{\hbar}{m\omega}}\). Using \(\hbar \approx 1.05 \times 10^{-34}\ \mathrm{J\cdot s}\), we have \(x_0 = \sqrt{\frac{1.05 \times 10^{-34}}{1.0 \times 10}} = \sqrt{1.05 \times 10^{-35}} \approx 3.24 \times 10^{-18}\ \mathrm{m}\).
- The classical amplitude is given by \(X_0 = \sqrt{2} |\alpha| x_0\). Substituting \(|\alpha| = 10\), we get \(X_0 = \sqrt{2} \times 10 \times 3.24 \times 10^{-18} \approx 4.58 \times 10^{-17}\ \mathrm{m}\).
- The uncertainty in position for a coherent state is constant and equal to the ground state uncertainty: \(\Delta x = \frac{x_0}{\sqrt{2}} = \sqrt{\frac{\hbar}{2m\omega}}\). Thus, \(\Delta x = \frac{3.24 \times 10^{-18}}{\sqrt{2}} \approx 2.29 \times 10^{-18}\ \mathrm{m}\).
- Calculate the ratio \(\frac{\Delta x}{X_0} = \frac{2.29 \times 10^{-18}}{4.58 \times 10^{-17}} = \frac{1}{\sqrt{2} |\alpha|} = \frac{1}{10\sqrt{2}} \approx 0.0707\).
Units check. Both \(\Delta x\) and \(X_0\) are lengths, so the ratio is dimensionless. The small value indicates that the uncertainty is about 7% of the amplitude, confirming that the state behaves classically to a good approximation.
Three quick tests
- Why do energy eigenstates \(|n\rangle\) not exhibit classical oscillatory motion in their probability density?
Answer
Because they are stationary states with time-independent probability densities \(|\psi_n(x)|^2\), whereas classical motion requires a time-dependent position distribution. - What is the defining property of a coherent state \(|\alpha\rangle\) in terms of ladder operators?
Answer
It is an eigenstate of the annihilation operator \(\hat{a}\), satisfying \(\hat{a}|\alpha\rangle = \alpha|\alpha\rangle\). - How does the width of a coherent state wave packet change with time?
Answer
It remains constant; coherent states do not spread, unlike generic superpositions of energy eigenstates.
Problem set for PU-202 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.
Derivations cited: D-010 · D-008 · Threads: matter, waves, energy