Question

which among is the correct statement for well behaved wave function in quantum mechanical system ?
1. \(\psi\) must be discontinuous
2. \(\psi\) should be squared-integrable.
3. \(\psi\) must be infinite everywhere
4. \(\psi\) should not be normalizable.

Answer 1

Correct Answer - Option 2 : \(\psi\) should be squared-integrable.

Concept:

• The wave quantity associated with a materialize particle(e.g. electrons) is\(\psi(x,t)\), we called it the wave function of the particle
• The quantity with which quantum mechanics is governed is the wave function\(\psi\) of a body.
• while \(\psi\) itself has no physical meaning but the square of its absolute magnitude \(|\psi|^2\) evaluated at a particular point at a particular time determine the probability of finding the particle at that instant
• The wave-function is usually complex, with both real and imaginary parts. 
• The probability density \(|\psi|^2\)needs to be positive, thus we take the complex conjugate of the given function to avoid the imaginary part. 

Explanation:​

• The wave-function for a realistic situation or well-behaved wave-function must satisfy the following condition

1. It should be finite everywhere to ensure that the probability of finding a particle in the given range is finite. 
2. the wave-function should be square-integrable, which means\(\int|\psi|^2 dx\) is finite. this ensures that the wave function can be normalized, making this integral equal to 1, the probability of finding the particle in whole space should be one 
3. It must be continuous everywhere, this must satisfy because \(|\psi|^2\) represent the probability density