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Principles of Quantum Mechanics

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Principles of Quantum Mechanics MATH3385/5385. Quantum Mechanics. Handout # 3: The Quantum Mechanics of Structureless Particles Principles of Quantum Mechanics Here we will study the principles of the QM of a structureless particle subject to a (possibly time-dependent) potential V (r,t). Structureless means that there are no other character- istics of the particle other than its position and momentum ( in contrast to e.g. particles with spin, a characteristic that we will introduce at a later stage). Our considerations con- cerning the wave packet description (see Handout # 2) has demonstrated that contrary to the classical perception of a particle (i.e. a point-like object which follows a geometric trajectory characterised at each point in time by its position and velocity) a fundamental uncertainty must be part of the new theory: in QM no longer can we measure both position as well as momentum of a particle simultaneously to arbitrary precision. This imprecision is not a fault of our apparatus or measuring device: it is fundamental to the theory. If we endeavour to measure the position, we lose track of the momentum and vice versa. We have to come to terms with this new and unfamiliar point of view on particles in the quantum theory. To capture both the particle aspect as well as the wave aspect, we need a description that achieves two things: • it allows for superposition of waves, thus incorporating the wave aspect, and • it has a built-in conservation law which accounts for the particle aspect (indeed: the particle is preserved; it is not supposed to vanish). We will now highlight the various ingredients of the theory. Wave function and probability interpretation The main idea behind quantum mechanics is to describe particles by a complex valued wave function ψ(r,t) which represents the spatial state of the particle. The word state is one of the key words in QM: instead of a particle having at each time t a precise value associated with it for the physical observables such as position and momentum, we say that the particle is prepared in a certain state described by the wave function at time t. What does the wave function represent physically? The interpretation (according to the Copenhagen school1) is the following: the absolute value P (r,t)= |ψ(r,t)|2 is a prob- ability density, meaning that 1This interpretation, which was promoted by N. Bohr and M. Born, has not remained undisputed. In fact, A. Einstein objected very strongly against the probability interpretation, saying that “God does not play dice”. Even today the discussions surrounding the probability interpretation of QM is carrying on. However, irrespective of the interpretation, the mathematical consistency of the theory is well-established. 1 |ψ(r,t)|2dr = the probability of finding a particle at time t in an infinitesimally small box of volume dr = dx dy dz located at r. This is quite a new and revolutionary idea: it introduces an amount of subjectiveness in the description of nature. The “observer” (i.e. the person or the device that performs a measurement), is no longer isolated from the object it observes (i.e. the particle), but influences in a fundamental way the state in which the particle is found. In fact, the act of performing a measurement actually prepares the particle in a given state, namely in the state where the measured quantity is relatively well-localised (the bulk of the wave packet lying within the error measurement from the value predicted). Thus, a measurement invokes an irreversible change in the state of the particle. However, moments after the measurement, the wave packet evolves and starts to spread as we lose track of the particle, and we need to perform another measurement to localise the particle once again (in either the momentum or position space). This procedure can be graphically represented as follows ↑ ↑ −→ measurement measurement time The quantum mechanical evolution of a particle: the dashed cloud represents the wave distribution of the particle in space, the fuzziness of which indicates the imprecision before measurement. A measurement localises the wave function, after which the wave packet spreads out, and another measurement is needed to pinch the wave packet again, etc. By the definition of probability, the total probability (i.e. the sum over all values of the relevant, i.e. measured, variables characterising the particle) must be equal to one. In the present case where the wave function is a function of the position we have: ∞ ∞ ∞ P (r,t) dr = 1 , dr = dx dy dz Z Z Z−∞ Z−∞ Z−∞ where the integral represents the integration over three-dimensional space. This expresses the fact that requires that the wave functions are square-integrable i.e. the integral |ψ(r,t)|2 dr < ∞ Z and we will work with wave functions that are normalised to one2. The matter in which function spaces ψ takes values is a rather subtle one and we leave this to later. 2 2 If the wave function is not normalised and we have R |ψ(r,t)| dr = C we can always redefine ψ by dividing by C1/2 such that it becomes normalised as long as ψ is square integrable. 2 The complexity of the wave function ψ allows us to explain the nontrivial interference patterns that were discussed in Handout # 2. Obviously, ψ itself is not a measurable quantity because in physics we can only measure real-valued objects. Thus, it seems that the phase factor exp(iθ) associated with the wave function ψ(r,t)= |ψ(r,t)|eiθ(r,t) is not of physical importance. This, however is an erroneous point of view because when iθ1 iθ2 one superimposes two waves ψ1 = |ψ1|e and ψ2 = |ψ2|e , the superimposed wave function does depend on the relative phase θ1 − θ2 of the waves. In fact, we have 2 2 iθ1 iθ2 |ψ1 + ψ2| = |ψ1|e + |ψ2|e 2 2 ∗ = |ψ1| + |ψ2| + 2Re( ψ1ψ2) 2 2 i(θ1−θ2) = |ψ1| + |ψ2| + 2|ψ1| |ψ2| Re e 2 2 = |ψ1| + |ψ2| + 2|ψ1| |ψ2| cos(θ1 − θ2) Note that the complexity of the wave function is essential in the appearance of the relative phase factor: we do not see it in the individual waves since |eiθ| = 1, but it appears in their interference. Evolution of the system and conservation law The main equation that governs the time evolution of the wave function is the Schr¨odinger equation (SE). We have already derived a simple case of the SE in Handout # 2 in the case of a one-dimenional free particle. In the general case of a particle moving in three- dimensional space subject to a potential V (r,t) (possibly time-dependent) we have the following equation: The Schr¨odinger Equation: ~ ∂ψ ~2 ∇2 ∇2 ∂2 ∂2 ∂2 i ∂t = − 2m ψ + V (r,t)ψ , = ∂x2 + ∂y2 + ∂z2 One important aspect of the SE is that it preserves probability. This is important, because the total probability should be a constant of the motion: once we know that the particle exists it will not disappear and thus we have certainty (i.e. probability equal to one) of finding the particle somewhere. This can easily seen as folows. Using the SE, and its complex conjugate, i.e. ∂ψ∗ ~2 −i~ = − ∇2ψ∗ + V (r,t)ψ∗ ∂t 2m we derive by multiplying the SE by ψ∗ and the above equation by ψ and then subtract: ∂ψ ∂ψ∗ ~2 i~ ψ∗ + ψ = − ψ∗∇2ψ − (∇2ψ∗)ψ ∂t ∂t 2m 3 where the terms containing the potential V cancel eachother. This can be rewritten as: ∂ ~2 i~ |ψ|2 = − ∇ · (ψ∗∇ψ − (∇ψ∗)ψ) ∂t 2m which leads to the conservation law: ∂ i~ P + ∇ · j = 0 , j = − (ψ∗∇ψ − (∇ψ∗)ψ) ∂t 2m and where as before P (r,t)= |ψ(r,t|2. The vector quantity j is called the probability cur- rent, whereas P is the probability density. The conservation law leads to the conservation of probability as follows: let us integrate the formula over a volume V (after which we let V grow infintely large so that we get an integral over the entire three-dimensional space). This leads to d P (r,t) dr = − ∇ · j dr = − j · dS dt ZV ZV ZS where we have used the divergence theorem (see MATH2360: vector calculus) to obtain an integral over the closed surface S that surrounds the volume V. Thus this integral depends only on the values of j (and hence of ψ and ψ∗) on the surface S. Since ψ is assumed to be quadratically integrable, and consequently ψ → 0 as |r|→∞ the integral on the right-hand side vanishes as we let the volume V grow. Thus, we obtain for the integrals over the entire space: d P (r,t) dr = 0 ⇒ P (r,t) dr = constant dt Z Z and consequently the total probability is preserved. This tells us that the Schr¨odinger equation is consistent with the normalisation of the total probability to one. Physical observables and expectation values In QM as in classical mechanics we want to be able to calculate the value of physical ob- servables. In classical mechanics this amounts to trying to predict the values of all relevant variables (position, momentum, etc.) by solving the equations of the motion, and nothing prevents us to do that in principle to arbitrary precision. In QM we can no longer do that: not all relevant variables can be simultaneously measured nor predicted to arbitrary precision: some physical quantities are incommensurable. If two observables (i.e. physical quantites that we can actually measure) are in principle simultaneously measurable then we call them commensurable and we can fully characterize the state of the particle by giving values to a large enough set of commensurable quantities.
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