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To demonstrate that if a wavefunction \( f(x) \) is a solution to the Schrödinger equation, then any constant times \( f(x) \) is also a solution, we need to rely on the fact that the Hamiltonian \( \hat{H} \) is a linear operator. Here's a step-by-step explanation of the proof using the concept of linearity.

The time-independent Schrödinger equation is given by:

\[
\hat{H} f(x) = E f(x)
\]

Where:
- \( \hat{H} \) is the Hamiltonian operator (linear and Hermitian),
- \( f(x) \) is a wavefunction,
- \( E \) is the energy eigenvalue corresponding to \( f(x) \).

Now, let's assume that \( f(x) \) is a solution to this equation. We need to show that any constant multiple of \( f(x) \), say \( g(x) = c f(x) \), where \( c \) is a constant, is also a solution.

### Step-by-step solution:

1. **Substitute** the wavefunction \( g(x) = c f(x) \) into the Schrödinger equation:

\[
\hat{H} g(x) = \hat{H} (c f(x))
\]

2. **Use the linearity of the Hamiltonian**:

Since \( \hat{H} \) is a linear operator, we can factor out the constant \( c \):

\[
\hat{H} (c f(x)) = c \hat{H} f(x)
\]

3. **Substitute the Schrödinger equation** for \( f(x) \) (i.e., \( \hat{H} f(x) = E f(x) \)):

\[
c \hat{H} f(x) = c E f(x)
\]

4. **Simplify the equation**:

\[
\hat{H} g(x) = E (c f(x)) = E g(x)
\]

Thus, \( g(x) = c f(x) \) is also a solution to the Schrödinger equation, with the same energy eigenvalue \( E \).

### Conclusion:

This shows that if \( f(x) \) is a solution to the Schrödinger equation, then any constant multiple of \( f(x) \) is also a solution, which follows from the linearity of the Hamiltonian operator.

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This completes the proof. Let me know if you'd like more details or if you want to see any part of this worked out in Mathematica!

### Here are some related questions you might find useful:
1. How does the Hermitian property of the Hamiltonian influence the solutions of the Schrödinger equation?
2. Can non-linear operators allow the same kind of scaling for wavefunctions as seen in this problem?
3. What are the physical implications of multiplying a wavefunction by a constant in quantum mechanics?
4. How does normalization of wavefunctions affect the solution when a constant is introduced?
5. Can the time-dependent Schrödinger equation also be solved similarly for a constant multiple of a solution?

### Tip:
When solving quantum mechanics problems, always ensure the linearity of the operators is preserved, as this property significantly simplifies the analysis of wavefunctions.

Demonstrate that if any wavefunction f(x) is a solution to the Schrödinger equation, then any constant times f(x) is also a solution. Use the fact that the Hamiltonian is a linear, Hermitian operator.

Learn how to prove that any constant multiple of a wavefunction solution is also a solution to the Schrödinger equation. This step-by-step solution covers key concepts in quantum mechanics, such as the linearity of the Hamiltonian and wavefunction properties, ideal for undergraduate physics students.

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