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Vishal Jain
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Vishal Jain
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A travelling wave in 3D can be represented as the following: $\vec{\Psi}(\vec{r},t) = \vec{A}e^{i(\vec{k} \cdot \vec{r}-\omega t)}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{k_x r_x -wt}$$\Psi_x = A_xe^{i(k_x r_x -wt)}$, whereas in reality, $\Psi_x = A_xe^{\vec{k} \cdot \vec{r}-wt}$$\Psi_x = A_xe^{i(\vec{k} \cdot \vec{r}-wt)}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

A travelling wave in 3D can be represented as the following: $\vec{\Psi}(\vec{r},t) = \vec{A}e^{i(\vec{k} \cdot \vec{r}-\omega t)}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{k_x r_x -wt}$, whereas in reality, $\Psi_x = A_xe^{\vec{k} \cdot \vec{r}-wt}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

A travelling wave in 3D can be represented as the following: $\vec{\Psi}(\vec{r},t) = \vec{A}e^{i(\vec{k} \cdot \vec{r}-\omega t)}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{i(k_x r_x -wt)}$, whereas in reality, $\Psi_x = A_xe^{i(\vec{k} \cdot \vec{r}-wt)}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

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Qmechanic
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A travelling wave in 3D can be represented as the following: $\Psi(\vec{r},t) = \vec{A}e^{\vec{k} \cdot \vec{r}-wt}$$\vec{\Psi}(\vec{r},t) = \vec{A}e^{i(\vec{k} \cdot \vec{r}-\omega t)}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{k_x r_x -wt}$, whereas in reality, $\Psi_x = A_xe^{\vec{k} \cdot \vec{r}-wt}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

A travelling wave in 3D can be represented as the following: $\Psi(\vec{r},t) = \vec{A}e^{\vec{k} \cdot \vec{r}-wt}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{k_x r_x -wt}$, whereas in reality, $\Psi_x = A_xe^{\vec{k} \cdot \vec{r}-wt}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

A travelling wave in 3D can be represented as the following: $\vec{\Psi}(\vec{r},t) = \vec{A}e^{i(\vec{k} \cdot \vec{r}-\omega t)}$. I’m not sure I fully understand this expression.

$\vec{A}$ to me means the amplitude vector, which gives the magnitude of the wave in different components of space eg $A_x,A_y,A_z$, in space described by cartesian coordinates. The exponential term I am assuming to be the phase term which tells us how the phase of the wave evolves with time.

My issue is that when writing out the x component of the above, I would’ve thought $\Psi_x = A_xe^{k_x r_x -wt}$, whereas in reality, $\Psi_x = A_xe^{\vec{k} \cdot \vec{r}-wt}$.

Can someone explain why we include the full wave vector in the exponent, rather than its component when describing any one component of the plane wave?

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Vishal Jain
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