Question

The wave functions for the ground and first excited states of a simple hartnonic oscillator are \(A e^{-b x^{2} / 2}\) and B.xe \(^{-b x^{2} / 2}\). Suppose you have two particles occupying these two states. (a) If distinguishable, an acceptable wave function would be \(A e^{-b x_{1}^{2} / 2} B x_{2} e^{-b x_{2}^{2} / 2}\). Calculate the probability that both particles would be on the positive side of the origin and divide by the total probability for both being found over all values of \(x_{1}\) and. \(x\), (This kind of nonnalizing-as-we-go will streamline things.) (b) Suppose now that the particles are indistin. guishable. Using the \(\pm\) symbol to reduce your work. calculate the same probability ratio, but assuming that their multiparticle wave function is either symmetric on antisymmetric. Comment on your results.

Short answer

The short answer would be the ratio \(P_{+} / P_{tot}\) for distinguishable case and two different quantities for the symmetric and antisymmetric indistinguishable cases. The comments on the result would depend on specific numerical results obtained from each calculation.

Step-by-step solution

Normalizing the wave functions

The wave functions provided need to be normalized. Normalizing implies that the total probability of finding the particle in all of space must be 1. Normalization of the wave functions should proceed as \( \int A^{2} e^{-b x^{2}} dx = 1 \) and \( \int B^{2} x^{2} e^{-b x^{2}} dx = 1 \).

Calculate the probability when particles are distinguishable

The wave function of this system is given as \(A e^{-b x_{1}^{2} / 2} B x_{2} e^{-b x_{2}^{2} / 2}\). To get the probability that both particles would be on the positive side of the origin, we need to square the wave function, and then integrate over all positive values of \(x_{1}\) and \(x_{2}\). We refer to this value as \(P_{+}\). We also need to find the total probability involving all values of \(x_{1}\) and \(x_{2}\), referred to as \(P_{tot}\). The required probability ratio is then \(P_{+} / P_{tot}\).

Calculation for indistinguishable particles

In the case of indistinguishable particles, the multi-particle wave function can either be symmetric or antisymmetric. The symmetric and antisymmetric wave function would be respectively given by: \((1/\sqrt{2})(\psi_1(x_1)\psi_2(x_2) + \psi_1(x_2)\psi_2(x_1))\) and \((1/\sqrt{2})(\psi_1(x_1)\psi_2(x_2) - \psi_1(x_2)\psi_2(x_1))\). We repeat the procedure in step 2 but with these new wave functions instead. Calculate the probability for positive and total values, then find the ratio \(P_{+}/P_{tot}\).

Comment on the results

Compare the obtained ratios for the distinguishable and indistinguishable cases. The conclusion would be different depending on whether the particles are distinguishable or indistinguishable, and also whether the wave function is symmetric or antisymmetric.

Key concepts

Harmonic Oscillator Wave Functions

The concept of wave functions in the context of the harmonic oscillator is crucial in quantum mechanics. A classical harmonic oscillator, such as a mass attached to a spring, has a well-known behavior characterized by oscillating back and forth around an equilibrium position. In quantum mechanics, however, this oscillator is described not by its position at a given time, but by a wave function that provides a probability distribution for where the particle might be found upon measurement.

The wave functions for the ground and first excited states of a quantum harmonic oscillator are represented mathematically by functions that resemble the Gaussian, or bell curve, shape. Specifically, the ground state wave function is given by a Gaussian function, symbolized as \(A e^{-b x^{2} / 2}\), while the first excited state has an additional factor of \(x\), represented as \(B x e^{-b x^{2} / 2}\).

These wave functions must be normalized to ensure that the probability of finding the particle somewhere in space is 1. Normalization is accomplished by determining the constants \(A\) and \(B\) through the integration of the squared wave function over all space. Once normalized, the wave function can be used to calculate various quantities, including the probability of finding particles in particular regions of space.

Probability in Quantum Systems

In quantum mechanics, the probability of finding a particle in a particular state or location is derived from the wave function. For a particle in the harmonic oscillator potential, this involves squaring the normalized wave function and integrating it over a specific range.

When dealing with two particles, one in the ground state and the other in the excited state, and counting them as distinguishable, we can simply multiply their individual wave functions to find the combined system's wave function. To find the probability that both particles are on the positive side of the origin, we square this combined wave function and integrate it over all positive values of \(x_1\) and \(x_2\). This integral gives us the probability \(P_+\).

Additionally, we must compare this to the total probability of finding both particles anywhere, given by \(P_{tot}\). The ratio \(P_+/P_{tot}\) then tells us the likelihood of our particular scenario occurring relative to all possible scenarios. Understanding this is fundamental in quantum mechanics as it directs how we interpret the behavior of systems at a microscopic level where classical intuitions no longer apply.

Symmetric and Antisymmetric Wave Functions

When the particles in a system are indistinguishable, as is often the case in quantum mechanics, we need to consider the principles of symmetry in their wave functions. The Pauli exclusion principle dictates that identical fermions (particles like electrons with half-integer spin) must have antisymmetric wave functions, whereas bosons (particles with integer spin) can have symmetric wave functions.

The symmetric wave function is essentially an average of the probability amplitudes for the two particles being in either state, signifying the indistinguishability of the particles. An antisymmetric wave function, on the other hand, includes a subtraction, which leads to a zero probability of finding both particles in the same state or location due to the exclusion principle.

In our exercise, the symmetric and antisymmetric wave functions are constructed by symmetrically or antisymmetrically combining the normalized wave functions of the two distinguishable particles. This results in unique probability distributions for finding the particles on either side of the origin. By calculating the ratio of probabilities for both particles being found on the positive side of the origin to the total probabilities, we can see how symmetry affects the outcomes for indistinguishable particles. This understanding extends to many quantum systems, from atoms to solid-state physics contexts, where symmetry plays an integral role in determining system properties.