Physics

Physics · modern-physics

Problem

(g) The theory of relativity instead posits that the energy of a particle is given by E=p2c2+m2c4E=\sqrt{p^{2} c^{2}+m^{2} c^{4}}. In accordance with this, we can try to guess a relativistic version of the Schrödinger equation: 1c2∂2ϕ∂t2−∂2ϕ∂x2+m2c2ℏ2ϕ=0(7)\frac{1}{c^{2}} \frac{\partial^{2} \phi}{\partial t^{2}}-\frac{\partial^{2} \phi}{\partial x^{2}}+\frac{m^{2} c^{2}}{\hbar^{2}} \phi=0 \tag{7} This is called the Klein-Gordon equation. Using the same guess as before, find ω\omega in terms of k\mathrm{k}. Hint: If you are careful, you should find that there is an infinite continuum of energy states extending down to negative infinity. This apparently mathematical issue hints at the existence of antimatter, and ultimately demonstrates to us that we must formulate quantum field theory to properly describe relativistic quantum physics. Given constants: 4. A complex dance In this problem, we will solve a number of differential equations corresponding to very different physical phenomena that are unified by the idea of oscillation. Oscillations are captured elegantly by extending our notion of numbers to include the imaginary unit number ii, strangely defined to obey i2=−1i^{2}=-1. In other words, rather than using real numbers, it is more convenient for us to work in terms of complex numbers. Exponentials are usually associated with rapid growth or decay. However, with the inclusion of complex numbers, imaginary "growth" and "decay" can be translated into oscillations by the Euler identity: eiθ=cos⁡θ+isin⁡θ(1)e^{i \theta}=\cos \theta+i \sin \theta \tag{1} Context question: (a) The usual form of Newton's second law (F⃗=ma⃗)(\vec{F}=m \vec{a}) breaks down when we go into a rotating frame, where both the centrifugal and Coriolis forces become important to account for. Newton's second law then takes the form F⃗=m(a⃗+2v⃗×Ω⃗+Ω⃗×(Ω⃗×r⃗))(2)\vec{F}=m(\vec{a}+2 \vec{v} \times \vec{\Omega}+\vec{\Omega} \times(\vec{\Omega} \times \vec{r})) \tag{2} For a particle free of forces confined to the x−yx-y plane in a frame which rotates about the zz axis with angular frequency Ω\Omega, this becomes the complicated-looking system of differential equations, 0=x¨+2Ωy˙−Ω2x0=x¨−2Ωx˙−Ω2y(3)\begin{aligned} & 0=\ddot{x}+2 \Omega \dot{y}-\Omega^{2} x \\ & 0=\ddot{x}-2 \Omega \dot{x}-\Omega^{2} y \end{aligned} \tag{3} where dots represent time derivatives. Defining η=x+iy\eta=x+i y, show that Equations 3 are equivalent to the following single (complex) equation: 0=η¨−2iΩη˙−Ω2η(4)0=\ddot{\eta}-2 i \Omega \dot{\eta}-\Omega^{2} \eta \tag{4} Context answer: 证明题 Context question: (b) Equation 4 is a version of the damped harmonic oscillator, and can be solved by guessing a solution η=αeλt\eta=\alpha e^{\lambda t}. Plugging in this guess, what must λ\lambda be? Context answer: λ=iΩ\lambda=i \Omega Context question: (c) Using your answer to part (b), and defining α=Aeiϕ\alpha=A e^{i \phi} where AA and ϕ\phi are real, find x(t)\mathbf{x}(\mathbf{t}) and y(t)\mathbf{y}(\mathbf{t}). This is the trajectory for a particle which is stationary with respect to the symmetry axis. While not required for this problem, an additional guess would reveal that η=βteλt\eta=\beta t e^{\lambda t} is also a solution. Context answer: x(t)=Acos⁡(Ωt+ϕ)x(t)=A \cos (\Omega t+\phi) , y(t)=Asin⁡(Ωt+ϕ)y(t)=A \sin (\Omega t+\phi) Context question: (d) The one-dimensional diffusion equation (also called the "heat equation") is given (for a free particle) by ∂ψ∂t=a∂2ψ∂x2(5)\frac{\partial \psi}{\partial t}=a \frac{\partial^{2} \psi}{\partial x^{2}} \tag{5} A spatial wave can be written as ∼eikx\sim e^{i k x} (larger kk 's correspond to waves oscillating on smaller length scales). Guessing a solution ψ(x,t)=Aeikx−iωt\psi(x, t)=A e^{i k x-i \omega t}, find ω\omega in terms of k. A relationship of this time is called a "dispersion relation." Context answer: ω=−ik2a\omega=-i k^{2} a Context question: (e) The most important equation of non-relativistic quantum mechanics is the Schrödinger equation, which is given by iℏ∂ψ∂t=−ℏ22m∂2ψ∂x2(6)i \hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2 m} \frac{\partial^{2} \psi}{\partial x^{2}} \tag{6} Using your answer to part (d), what is the dispersion relation of the Schrödinger equation? Context answer: ω=ℏk22m\omega=\frac{\hbar k^{2}}{2 m} Context question: (f) If the energy of a wave is E=ℏωE=\hbar \omega and the momentum is p=ℏkp=\hbar k, show that the dispersion relation found in part (e) resembles the classical expectation for the kinetic energy of a particle, E=mv2/2\mathrm{E}=\mathrm{mv}^{2} / \mathbf{2}. Context answer: 证明题

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