# Where is a particle most likely to be? (Griffiths Quantum Mechanics)

• blackbeans
In summary, the conversation discusses the possibility of a wave function being non-differentiable at certain points and how it relates to physically realizable states. The question of finding the coordinate where the particle is most likely to be is also raised, and the methods for calculating the maximum of a wave function are discussed. It is noted that while it is ideal for the wave function to be differentiable, it is not a strict requirement.
blackbeans
Homework Statement
Hi there, it seems more convenient to post a picture of the problem in question. More specifically, problem 1(c).
Relevant Equations
the schrodinger equation.
The wave function described seems impossible. Wave functions have to be differentiable at all points, right? Otherwise they don't represent a physically realizable state. The wave function in the example isn't differentiable at x=A, the maximum point. Also, for problem (c), I know it's visually simple to see the answer, but for a more general case, how would i find the coordinate where the "particle is most likely to be"? Would I take the derivative of |psi|^2 or |psi| to find the absolute maxima? Do they provide the same result? Is there a simpler method?

Theoretically, wave functions need only be square integrable. You could, however, look at this sort of function an idealised approximation to a function that would have a differentiable maximum.

As you state, the modulus squared of the wave function represents the PDF of the particle's position. You calculate the maximum as you would for any function, using calculus or otherwise.

blackbeans
I see. I just assumed that the wave function had to be differentiable everywhere, since its derivative shows up in the Schrodinger Eq. Thank you!

blackbeans said:
I see. I just assumed that the wave function had to be differentiable everywhere, since its derivative shows up in the Schrodinger Eq. Thank you!
Technically it's better if it is differentiable. But differentiable almost everywhere is probably good enough

## 1. Where is a particle most likely to be located in a quantum system?

The location of a particle in a quantum system cannot be predicted with certainty, according to the principles of quantum mechanics. Instead, the particle's location is described by a probability distribution that indicates where the particle is most likely to be found. This distribution is determined by the wave function of the particle.

## 2. Can a particle be in multiple places at once?

According to the principles of quantum mechanics, a particle can exist in a state of superposition, meaning it can be in multiple places or states simultaneously. However, upon measurement, the particle will collapse into a specific location or state, as determined by the probability distribution described by its wave function.

## 3. How does the uncertainty principle affect the location of a particle?

The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle simultaneously. This means that the more precisely we know the position of a particle, the less precisely we can know its momentum, and vice versa. Therefore, the exact location of a particle can never be known with certainty.

## 4. Can a particle's location be influenced by external factors?

Yes, a particle's location can be influenced by external factors such as other particles or fields. This is known as quantum entanglement, where the state of one particle can affect the state of another particle, regardless of their distance apart. This can result in a change in the probability distribution and therefore the most likely location of the particle.

## 5. How do we measure the location of a particle in a quantum system?

We can measure the location of a particle in a quantum system by using a physical measurement tool, such as a position detector. However, due to the uncertainty principle, the measurement will only give us a range of possible positions where the particle may be located, rather than an exact location. The precise location of the particle can only be determined by the wave function and its associated probability distribution.

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