Question

A particle is described by $$ \psi(x)=\left\\{\begin{array}{ll} C & |x| \leq+\frac{1}{2} w \\ 0 & |x|>\frac{1}{2} w \end{array}\right. $$ What momenta can never be measured?

Short answer

The momenta that can never be measured are associated with the positions outside the region \( |x| \leq \frac{1}{2} w \) where the wave function is zero. To obtain specific momenta values, however, values of constants \( C \) and \( w \) are required.

Step-by-step solution

Identify the region of zero wave function

From the given wave function, \( \psi(x) \), it's observed that the wave function becomes zero when \( |x| > \frac{1}{2} w \). Therefore, the region of zero probability of finding the particle is reported.

Understand the implications of a zero wave function

A zero wave function implies that the probability of finding the particle at those positions is zero, and thus, the momenta associated can never be measured.

Apply Fourier Transform

To find the possible momenta values, apply Fourier Transform to the wave function. The Fourier Transform will provide the probability distribution for the momenta by transferring to momentum space. Here, it is important to interpret that the constant wave function \( \psi(x) = C \) inside the region \( |x| \leq \frac{1}{2} w \) indicates there is equal probability to measure each value of momenta for the particle inside this region. Similarly, zero wave function for \( |x| > \frac{1}{2} w \) means zero probability for certain momentum values.

Conclusion

In conclusion, given that the wave function is zero for \( |x| > \frac{1}{2} w \), this implies the inability to measure these momenta. The specific values of these unmeasurable momenta depend on the values of the constant \( C \) and \( w \), which have not been specified in this particular exercise. Without these values, providing a detailed range or exact values of momenta is not possible.

Key concepts

Wave Function

In quantum mechanics, the wave function, typically represented by \( \psi(x) \), is fundamental in describing the state of a particle. It is essentially a mathematical description that provides all the information about the particle's state.

The wave function can be complex, and its square modulus, \( |\psi(x)|^2 \), gives the probability density of finding the particle at a position \( x \). In simple terms, it tells us where the particle is likely to be found. In the given exercise, the wave function \( \psi(x) = C \) is non-zero only within a defined region \(-\frac{1}{2}w \leq x \leq \frac{1}{2}w\). This suggests that:

• The particle is equally likely to be located anywhere within this region.
• Outside this region, the probability of finding the particle drops to zero, as the wave function is zero.

Understanding these "allowed" and "forbidden" regions aids in knowing where the particle might be found based on the wave function's configuration.

Fourier Transform

The Fourier Transform is a critical tool in quantum mechanics used to transition between different representations of a system, particularly from position space to momentum space.

In the context of wave functions, performing a Fourier Transform on \( \psi(x) \) provides the momentum space wave function. This new representation, usually denoted as \( \phi(p) \), then tells us about the momentum distribution of the particle. Key points include:

• This transformed function informs us which momentum values are more likely and which are unlikely or impossible.
• The region where the original wave function is zero (\(|x| > \frac{1}{2}w\)) leads to corresponding zero probability densities in momentum space for certain momentum values.

By understanding Fourier Transforms, we comprehend the dual nature of particles, as both their position and momentum distributions are necessary for a complete description.

Momentum

Momentum in quantum mechanics is tied intricately to the wave function via the Fourier Transform. Unlike in classical mechanics, where momentum is definitively calculated, quantum momentum is probabilistically distributed.

In the presented exercise, momentum values can be understood by analyzing the transformed \( \phi(p) \). Note that:

• Regions of zero probability in real space translate to zero probability for particular momentum values.
• These unmeasurable momenta do not contribute to the observable properties of the system, indicating constraints set by the wave function's form.

The concept of momentum here reveals that precise measurements in quantum systems involve observing distribution patterns rather than fixed values. When dealing with wave functions, gaining insights into momentum enhances our overall perception of quantum systems.