QuestionJune 6, 2025

Which of the following describe the expansion coefficients for a general state? a_(n)=langle psi _(n)vert psi rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi (x) a_(n)=(sum _(n)vert psi _(n)rangle )vert psi rangle a_(n)=langle psi _(n)vert psi rangle ^ast =int _(-infty )^infty dxpsi _(n)(x)psi ^ast (x) a_(n)=langle psi _(n)vert psi _(m)rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi _(m)(x)

Which of the following describe the expansion coefficients for a general state? a_(n)=langle psi _(n)vert psi rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi (x) a_(n)=(sum _(n)vert psi _(n)rangle )vert psi rangle a_(n)=langle psi _(n)vert psi rangle ^ast =int _(-infty )^infty dxpsi _(n)(x)psi ^ast (x) a_(n)=langle psi _(n)vert psi _(m)rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi _(m)(x)
Which of the following describe the expansion coefficients for a general state? a_(n)=langle psi _(n)vert psi rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi (x) a_(n)=(sum _(n)vert psi _(n)rangle )vert psi rangle a_(n)=langle psi _(n)vert psi rangle ^ast =int _(-infty )^infty dxpsi _(n)(x)psi ^ast (x) a_(n)=langle psi _(n)vert psi _(m)rangle =int _(-infty )^infty dxpsi _(n)^ast (x)psi _(m)(x)

Solution
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Answer

a_{n}=\langle \psi _{n}\vert \psi \rangle =\int _{-\infty }^{\infty }dx\psi _{n}^{\ast }(x)\psi (x) Explanation 1. Identify the correct formula for expansion coefficients The expansion coefficient a_n for a general state \vert \psi \rangle in terms of basis states \vert \psi_n \rangle is given by the inner product a_n = \langle \psi_n \vert \psi \rangle. This corresponds to the integral form a_n = \int_{-\infty}^{\infty} dx \, \psi_n^*(x) \psi(x). 2. Evaluate each option - Option 1: a_{n}=\langle \psi _{n}\vert \psi \rangle =\int _{-\infty }^{\infty }dx\psi _{n}^{\ast }(x)\psi (x) is correct. - Option 2: a_{n}=(\sum _{n}\vert \psi _{n}\rangle )\vert \psi \rangle is incorrect as it does not represent an inner product. - Option 3: a_{n}=\langle \psi _{n}\vert \psi \rangle ^{\ast }=\int _{-\infty }^{\infty }dx\psi _{n}(x)\psi ^{\ast }(x) is incorrect because it takes the complex conjugate of the correct expression. - Option 4: a_{n}=\langle \psi _{n}\vert \psi _{m}\rangle =\int _{-\infty }^{\infty }dx\psi _{n}^{\ast }(x)\psi _{m}(x) is incorrect as it involves two different states \vert \psi_n \rangle and \vert \psi_m \rangle.

Explanation

1. Identify the correct formula for expansion coefficients<br /> The expansion coefficient $a_n$ for a general state $\vert \psi \rangle$ in terms of basis states $\vert \psi_n \rangle$ is given by the inner product $a_n = \langle \psi_n \vert \psi \rangle$. This corresponds to the integral form $a_n = \int_{-\infty}^{\infty} dx \, \psi_n^*(x) \psi(x)$.<br /><br />2. Evaluate each option<br /> - Option 1: $a_{n}=\langle \psi _{n}\vert \psi \rangle =\int _{-\infty }^{\infty }dx\psi _{n}^{\ast }(x)\psi (x)$ is correct.<br /> - Option 2: $a_{n}=(\sum _{n}\vert \psi _{n}\rangle )\vert \psi \rangle$ is incorrect as it does not represent an inner product.<br /> - Option 3: $a_{n}=\langle \psi _{n}\vert \psi \rangle ^{\ast }=\int _{-\infty }^{\infty }dx\psi _{n}(x)\psi ^{\ast }(x)$ is incorrect because it takes the complex conjugate of the correct expression.<br /> - Option 4: $a_{n}=\langle \psi _{n}\vert \psi _{m}\rangle =\int _{-\infty }^{\infty }dx\psi _{n}^{\ast }(x)\psi _{m}(x)$ is incorrect as it involves two different states $\vert \psi_n \rangle$ and $\vert \psi_m \rangle$.
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