Does a Sudden Change in Hamiltonian Affect Wave Function Continuity?

[mar0]

Homework Statement

The Hamiltonian of a quantum system suddenly changes by a finite amount. Show that the wave function must change continuously if the time-dependent Schrodinger equation is to be valid throughout the change.

Homework Equations

Time independent Schrodinger equation and Hamiltonian;

The Attempt at a Solution

I don't understand this question at all. Maybe its those words. Please some help!
And does this contradict classical physics?

 

[mark726]

Thank you for your post. I understand that this question may seem confusing at first, but let me try to break it down for you.

The Hamiltonian is a mathematical operator that represents the total energy of a quantum system. It is a key component of the time-dependent Schrodinger equation, which describes how the wave function of a quantum system changes over time.

Now, the question is asking you to show that if the Hamiltonian suddenly changes by a finite amount, the wave function must also change continuously in order for the time-dependent Schrodinger equation to still be valid. In other words, the wave function cannot have any sudden jumps or discontinuities during the change in the Hamiltonian. This is because the Schrodinger equation is based on the principle of conservation of energy, and any sudden changes in the Hamiltonian would violate this principle.

As for your question about classical physics, the answer is yes and no. In classical physics, the concept of wave function and its continuous evolution over time does not exist. However, the principle of conservation of energy still applies, so in that sense, the idea of a continuous change in the wave function is not contradictory to classical physics.

I hope this helps clarify the question for you. If you need further assistance, please don't hesitate to ask. Good luck with your studies!