# Why should wave function be continuous?

• quanjia
In summary, the conversation discusses the continuity of wave functions and their relationship to probability. It is noted that while wave functions can be complex, their absolute value, which represents probability, should be continuous. The conversation also touches on the importance of continuity for the solutions of the Schrodinger equation, and how it pertains to the concept of delta potential. The reason for considering the wave function continuous at x=0, despite the discontinuity in its derivative, is also discussed.
quanjia
I think it needn't be continuous even if the probability should.
Wave function can be complex while probability is its absolute value.

quanjia said:
I think it needn't be continuous even if the probability should.
Wave function can be complex while probability is its absolute value.

Sure, and if a complex function has a singularity, what is its absolute value there?

I'm sorry that I can't catch you whole meaning exactly.

Let me construct an example as follow (f(x) is the wave function over x):
When 0<x<1/2, f(x)=e^(i*x),
when 1/2<x<1, f(x)= -2x+2,
and otherwise, f(x)=0.
there is only one singularity at x=1/2 with the absolute value is 1.

By the way,I don't know how to paste a picture or a formula quickly in this forum.How could I get information about that?

There are at least two problems here: there's no Hamiltonian for one. But more serious, in the function above, obviously the limit from the right and the limit from the left at x=1/2 don't agree. the only way this could happen physically is with some kind of weird barrier at x=1/2. The basic Schrodinger Eq. only allows discontinuous solutions for disjoint regions; that is continuity within a region, but not region-to-region. This, of course, is a well known aspect of differential equations.

Physically it will take a large perturbation to shift the absolute value of by a substantial amount over a very short region of space. So, it makes sense to me to extrapolate, and suggest that an amost infinite potential change is required to make an almost discontinuous change in a wave function. That's the intuitive reason why discontinuous wave functions are troublesome, and really refer to differences between two or more disjoint regions of space.
Regards,
Reilly Atkinson

One of the more important reasons that the wave function needs to be continuous is that

$$\hat{p} \equiv - i \hslash \nabla$$

so what happens to the expectation value of momentum if you have a discontinuous wave function?

Also, since the hamiltonian is a partial differential equation that is linear, there are a lot of theorems, one of which pretty much requires that the solutions be continuous and twice-differentiable.

Thank you,reilly and StatMechGuy.

But when we deal with delta potential,why we consider the wave function continuous at x=0 where the potential is infinite and the differential of the wave function is discontinuous?
Is it just for simplity in analyse and calculate.

Since the wave function should be a solution of the Schrödinger equation, it must be differentiable, thus also be continuous.

quanjia said:
Thank you,reilly and StatMechGuy.

But when we deal with delta potential,why we consider the wave function continuous at x=0 where the potential is infinite and the differential of the wave function is discontinuous?
Is it just for simplity in analyse and calculate.

You can go back to the Schrodinger equation and then integrate immediately around the delta function. If you assume the wave function is continuous, you get for sufficiently crazy potentials that the derivative of the wave function is discontinuous. That said, you never actually see a delta function potential in nature.

## 1. What is a wave function and why is it important?

A wave function is a mathematical representation used in quantum mechanics to describe the behavior of particles. It is important because it allows us to predict the probability of finding a particle in a certain state or location.

## 2. Why should a wave function be continuous?

A continuous wave function means that there are no sudden jumps or discontinuities in the behavior of the particle. This is important because it ensures that the predictions made by the wave function are physically meaningful and accurate.

## 3. How is the continuity of a wave function related to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. A continuous wave function helps to satisfy this principle by showing that there are no abrupt changes in the particle's position or momentum.

## 4. Can a wave function ever be discontinuous?

In principle, yes, a wave function can be discontinuous. However, this would result in physically meaningless predictions and is therefore not preferred. In most cases, a continuous wave function is more accurate and useful.

## 5. What happens if a wave function is not continuous?

If a wave function is not continuous, it means that there are abrupt changes in the particle's behavior, which would result in inaccurate predictions. This could lead to contradictions and inconsistencies in the laws of quantum mechanics.

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