Sketch a plot of the wavefunction ψ=sinn x for the domain 0 ≤x ≤πfor n=1,2, and 3 . What is the

step 1 In this question we've been asked to determine or rather to plot the wave functions for sine nx,

Sketch a plot of the wavefunction ψ=sinn x for the domain 0...

Sketch a plot of the wavefunction $\psi=\sin n x$ for the domain $0 \leq x \leq \pi$ for $n=1,2,$ and $3 .$ What is the relationship between the "quantum number" $n$ and the number of nodes in the wavefunction?

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Key Concepts

Quantum Number

In quantum mechanics, a quantum number is a discrete value that characterizes a specific state of a system. It often determines properties such as energy, angular momentum, or the number of nodes in the associated wavefunction. In many problems, especially those involving bound states, the quantum number helps to enumerate the different allowed states of the system.

Nodes

Nodes are specific points or regions in the spatial domain where the wavefunction exactly becomes zero. They serve as indicators of the state’s complexity—more nodes generally correspond to higher energy states. In many one-dimensional quantum systems, the number of interior nodes (excluding fixed boundaries where the function might be zero due to boundary conditions) increases systematically with the quantum number.

Relationship Between Quantum Number and Nodes

For wavefunctions defined by sinusoidal functions, such as those in a particle-in-a-box model, the relationship between the quantum number and nodes is typically straightforward: the number of internal nodes is equal to one less than the quantum number (i.e., n-1). This reflects the fact that the ground state has no interior nodes, while each subsequent state contains one additional node, representing an increased level of excitation.

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