Ashcroft Mermin eng · Problem 6

The de Haas-van Alphen Effect

Physics — Statistical Mechanics · semiclassical-electric-field-bloch-theorem

Problem

The validity of the semiclassical result k(t)=k(0)−eEt/ℏ\mathbf{k}(t) = \mathbf{k}(0) - e\mathbf{E}t/\hbar for an electron in a uniform electric field is strongly supported by the following theorem (which also provides a useful starting point for a rigorous theory of electric breakdown): Consider the time-dependent Schr\"odinger equation for an electron in a periodic potential U(r)U(\mathbf{r}) and a uniform electric field: iℏ∂ψ∂t=[−ℏ22m∇2+U(r)+eE⋅r]ψ=Hψ.(12.78)i\hbar \frac{\partial \psi}{\partial t} = \left[-\frac{\hbar^2}{2m}\nabla^2 + U(\mathbf{r}) + e\mathbf{E} \cdot \mathbf{r}\right] \psi = H\psi. \tag{12.78} Suppose that at time t=0t = 0, ψ(r,0)\psi(\mathbf{r}, 0) is a linear combination of Bloch levels, all of which have the same wave vector k\mathbf{k}. Then at time tt, ψ(r,t)\psi(\mathbf{r}, t) will be a linear combination of Bloch levels, all of which have the wave vector k−eEt/ℏ\mathbf{k} - e\mathbf{E}t/\hbar. Prove this theorem by noting that the formal solution to the Schr\"odinger equation is ψ(r,t)=e−iHt/ℏψ(r,0),(12.79)\psi(\mathbf{r}, t) = e^{-iHt/\hbar} \psi(\mathbf{r}, 0), \tag{12.79} and by expressing the assumed property of the initial level and the property to be proved of the final level in terms of the effect on the wave function of translations through Bravais lattice vectors.
Intuitive explanation

The key insight is that the electric field term eE⋅re\mathbf{E} \cdot \mathbf{r} acts like a time-dependent shift of the crystal momentum. You'll show this by proving that if your initial state respects a certain translation symmetry (characteristic of Bloch states with wave vector k\mathbf{k}), then the time-evolved state respects the same translation symmetry but with k\mathbf{k} replaced by k−eEt/ℏ\mathbf{k} - e\mathbf{E}t/\hbar. The trick is to construct a unitary transformation that "absorbs" the field's effect into a redefinition of the wave vector, leaving the periodic structure of the Hamiltonian intact.

Intuitive explanation

The key insight is that the electric field term eE⋅re\mathbf{E} \cdot \mathbf{r} acts like a time-dependent shift of the crystal momentum. You'll show this by proving that if your initial state respects a certain translation symmetry (characteristic of Bloch states with wave vector k\mathbf{k}), then the time-evolved state respects the same translation symmetry but with k\mathbf{k} replaced by k−eEt/ℏ\mathbf{k} - e\mathbf{E}t/\hbar. The trick is to construct a unitary transformation that "absorbs" the field's effect into a redefinition of the wave vector, leaving the periodic structure of the Hamiltonian intact.

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