According to quantum mechanics, the x

According to quantum mechanics, the $x$ - and $y$ -coordinates of a particle confined to the region $\mathcal{R}=[0,1] \times[0,1]$ are random variables with joint probability density function $$p(x, y)=\left\{\begin{array}{ll}{C \sin ^{2}(2 \pi \ell x) \sin ^{2}(2 \pi n y)} & {\text { if }(x, y) \in \mathcal{R}} \\ {0} & {\text { otherwise }}\end{array}\right.$$ The integers $\ell$ and $n$ determine the energy of the particle, and $C$ is a constant. (a) Find the constant $C .$ (b) Calculate the probability that a particle with $\ell=2, n=3$ lies in the region $\left[0, \frac{1}{4}\right] \times\left[0, \frac{1}{8}\right]$

According to quantum mechanics, the $x$ - and $y$ -coordinates of a particle confined to the region $\mathcal{R}=[0,1] \times[0,1]$ are random variables with joint probability density function
$$p(x, y)=\left\{\begin{array}{ll}{C \sin ^{2}(2 \pi \ell x) \sin ^{2}(2 \pi n y)} & {\text { if }(x, y) \in \mathcal{R}} \\ {0} & {\text { otherwise }}\end{array}\right.$$
The integers $\ell$ and $n$ determine the energy of the particle, and $C$ is a constant.
(a) Find the constant $C .$
(b) Calculate the probability that a particle with $\ell=2, n=3$ lies in the region $\left[0, \frac{1}{4}\right] \times\left[0, \frac{1}{8}\right]$

Key Concepts

Probability Calculations in Confined Quantum Systems

In quantum mechanics, when a particle is confined to a specific region, its spatial probability distribution is determined by its wavefunction. Calculating the probability of finding a particle in a particular region involves integrating the probability density over that region. This process is fundamental for predicting outcomes in experiments and understanding phenomena such as quantization of energy levels in confined systems.

Normalization of Probability Density Functions

In quantum mechanics and probability theory, any probability density function must be normalized so that the total probability over the entire space is equal to one. This requires integrating the density function over its domain and solving for any constant factors that enforce this condition. Normalization is essential not only for the mathematical consistency of the theory but also for making meaningful physical predictions.

Separation of Variables in Joint Distributions

When a joint probability density function can be expressed as a product of functions, each depending on a single variable, the concept of separation of variables is employed. This technique simplifies the integration process by allowing the multidimensional integral to be broken down into a product of one-dimensional integrals. This approach is particularly useful in quantum mechanics where separable wavefunctions describe systems with independent degrees of freedom.

Trigonometric Integrals

Trigonometric integrals, such as those involving sinusoidal functions squared, are common in quantum mechanics due to the wave nature of particles. The evaluation of integrals of functions like sin^2 over a period or a fraction thereof is crucial for normalizing wavefunctions and computing probabilities. Understanding these integrals is important as they appear in both the normalization conditions and the calculation of observables in quantum systems.

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