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At first let's take a look at the eigenvalue problem for the momentum operator in x-representation.

[itex]

-i \hbar \frac{d}{dx} \psi(x)=p \psi(x) \Rightarrow \psi_p(x)=C e^{{ipx}/{\hbar}}

[/itex]

The orthogonality condition is:

[itex]

\langle p_i|p_j \rangle=C^2 \int_R e^{i(p_j-p_i)x/{\hbar}} dx=C^2 \delta(p_i-p_j)

[/itex]

And so the spectrum of the momentum operator is continuous

Now consider the particle in infinite well as an example.

The energy levels are given by:

[itex]

E_n=\large{\frac{n^2 h^2}{8 m L^2}}

[/itex]

We can find the

[itex]

\large{\frac{p_n^2}{2m}}\small{=E_n} \Rightarrow \large{\frac{p_n^2}{2m}\small{=}\frac{n^2 h^2}{8 m L^2}} \small{\Rightarrow p_n^2}=\large{\frac{n^2 h^2}{4 L^2}} \small{\Rightarrow p_n}=\large{\frac{n h}{2L}}

[/itex]

Which means the momentum is quantized in this example.But we know that the spectrum of the momentum operator should always be continuous.

What am I missing?

Thanks

[itex]

-i \hbar \frac{d}{dx} \psi(x)=p \psi(x) \Rightarrow \psi_p(x)=C e^{{ipx}/{\hbar}}

[/itex]

The orthogonality condition is:

[itex]

\langle p_i|p_j \rangle=C^2 \int_R e^{i(p_j-p_i)x/{\hbar}} dx=C^2 \delta(p_i-p_j)

[/itex]

And so the spectrum of the momentum operator is continuous

**regardless of the problem at hand.**Now consider the particle in infinite well as an example.

The energy levels are given by:

[itex]

E_n=\large{\frac{n^2 h^2}{8 m L^2}}

[/itex]

We can find the

*momentum levels*:[itex]

\large{\frac{p_n^2}{2m}}\small{=E_n} \Rightarrow \large{\frac{p_n^2}{2m}\small{=}\frac{n^2 h^2}{8 m L^2}} \small{\Rightarrow p_n^2}=\large{\frac{n^2 h^2}{4 L^2}} \small{\Rightarrow p_n}=\large{\frac{n h}{2L}}

[/itex]

Which means the momentum is quantized in this example.But we know that the spectrum of the momentum operator should always be continuous.

What am I missing?

Thanks

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