# Assuming Wave Function Collapse Actually Exists, Can a Wavefunction Collapse into Another Wavefunction that is not the Delta Function

Suppose we have a time independent potential and suppose $$\psi_1(x)$$ and $$\psi_2(x)$$ are two stationary states of the potential with energies $$E_1$$ and $$E_2$$. Suppose the wavefunction is $$\Psi(x,t) = \frac{1}{\sqrt{2}} \bigg(\psi_1 e^{-iE_1 t/\hbar} + \psi_2 e^{-iE_2 t/\hbar}\bigg)$$

Scenario 1: Suppose at a particular time $$t_0$$ I measure the position. The theoretical probability I get a position between $$a$$ and $$b$$ is $$\int_a^b |\Psi(x,t_0)|^2 dx$$

Scenario 2: Suppose at a time $$t < t_0$$ I measure the energy and obtain $$E_1$$. Then suppose at time $$t_0$$ I measure the position. What would be the theoretical probability that I get a position between $$a$$ and $$b$$? Will it be
$$\int_a^b |\Psi(x,t_0)|^2 dx$$ once again, or will it be

$$\int_a^b |\psi_1(x)|^2 dx?$$

• "Can a Wavefunction Collapse into Another Wavefunction that is not the Delta Function?" Answer: Yes.
– hft
Commented Dec 9, 2022 at 23:17

According to the Copenhagen interpretation of quantum mechanics, measuring energy $$E_1$$ can be thought of as a projection/collapse of your initial superposition state onto the energy eigenstate $$\psi_1$$, caused by the measurement.
So if you measure eigenvalue $$E_1$$ at $$t, then your results for measurements at $$t\geq t_0$$ will be given by
$$\int^b_a |\psi_1(x)|^2dx.$$