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Why is energy expressed as the second-order partial differential of a wave function in quantum mechanics?
You are referring to the Schrodinger Equation. This is because it comes from the classical view that the total energy is equal to the hamiltonian of a system:
Kinetic Energy + Potential Energy = Total energy.
Classically the kinetic energy is (1/2)mv2 = p2/(2m) ; where m is mass, v is velocity, p is momentum (p=mv).
Now the momentum operator in QM is p=iħ∇ ;where ∇ is the gradient operator.
This therefore yields the QM hamiltonian [-ħ2∇2/(2m) + V(x,y,z)]Ψ = EΨ
Now a more fun question to ask would be "Why is the Hamiltonian a function of the second-order partial differential with respect to position but the time dependent is only a function of a first-order differential with respect to time?"
meaning
HΨ = -iħ(dΨ/dt) or
[-ħ2∇2/(2m) + V(x,y,z)]Ψ = -iħ(dΨ/dt)
hint: Think Maxwell's Equations!
What else can I help you with?
What are the Differentiate the sine wave and cosine wave?
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
What is the the function of a differential?
your mom idiot
What is the difference between differential and derivative in the field of calculus?
The derivative refers to the rate at which a function changes with respect to another measure. The differential refers to the actual change in a function across a parameter. The differential of a function is equal to its derivative multiplied by the differential of the independent variable . The derivative of a function is the the LIMIT of the ratio of the increment of a function to the increment of the independent variable as the latter tends to zero.
What is the purpose of differential calculus?
Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
What is the difference between differential and differentiation?
A differential is the result gained when mathematical differentiation is applied to a function. Differentiation in maths is the function which finds the gradient of a function in terms of x. Differentiation in biology is the specialisation of unspecialised cells such as stem cells into active cells.
How is the delta function used in quantum mechanics?
The delta function is used in quantum mechanics to represent a point-like potential or a point-like particle. It is often used in solving differential equations and describing interactions between particles in quantum systems.
How is the momentum operator derivation performed in quantum mechanics?
In quantum mechanics, the momentum operator derivation is performed by applying the principles of wave mechanics to the momentum of a particle. The momentum operator is derived by considering the wave function of a particle and applying the differential operator for momentum. This operator is represented by the gradient of the wave function, which gives the direction and magnitude of the momentum of the particle.
What are the Differentiate the sine wave and cosine wave?
The differential of the sine function is the cosine function while the differential of the cosine function is the negative of the sine function.
What is the the function of a differential?
your mom idiot
What is the difference between differential and derivative in the field of calculus?
The derivative refers to the rate at which a function changes with respect to another measure. The differential refers to the actual change in a function across a parameter. The differential of a function is equal to its derivative multiplied by the differential of the independent variable . The derivative of a function is the the LIMIT of the ratio of the increment of a function to the increment of the independent variable as the latter tends to zero.
What is the function of differential in a vehicle?
your mom idiot
What two fields of study do the principles of bio-mechanics come from?
The principles of bio-mechanics come from the fields of biology and mechanics. Bio-mechanics applies the principles of mechanics to understand how living organisms move and function.
What is the definition of the wave function in quantum mechanics?
In quantum mechanics, the wave function is a mathematical function that describes the behavior of a particle or system of particles. It represents the probability amplitude of finding a particle in a particular state or position.
What is the purpose of differential calculus?
Differential Calculus is to take the derivative of the function. It is important as it can be applied and supports other branches of science. For ex, If you have a velocity function, you can get its acceleration function by taking its derivative, same relationship as well with area and volume formulas.
What is the difference between differential and differentiation?
A differential is the result gained when mathematical differentiation is applied to a function. Differentiation in maths is the function which finds the gradient of a function in terms of x. Differentiation in biology is the specialisation of unspecialised cells such as stem cells into active cells.
What is the function of belts in the mechanics of machines?
The function of belts is to transmit power from one place to other in a machine.
What is the significance of the psi star function in quantum mechanics?
The psi star function in quantum mechanics is significant because it represents the complex conjugate of the wave function, allowing us to calculate probabilities and observables in quantum systems. It helps us understand the behavior of particles at the quantum level and is essential for making predictions in quantum mechanics.
