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Quantum Spin Calculator

Calculate quantum spin properties including spin angular momentum, magnetic moment, and spin projections

Parameters

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Tⓘ
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Calculated Values

Spin Magnitude:
9.1329e−35;J⋅s9.1329e-35;J·s
Spin Projection:
5.2729e−35;J⋅s5.2729e-35;J·s
Magnetic Moment:
−9.2848e−24;J/T-9.2848e-24;J/T
Energy in Field:
9.2848e−24;J9.2848e-24;J
Possible Projections:
2.00;2.00;

Visualization

Quantum Spin

Quantum spin is an intrinsic form of angular momentum carried by elementary particles, composite particles, and atomic nuclei. Unlike classical angular momentum, spin is quantized and has no classical analogue.

The spin quantum number s determines the magnitude of spin angular momentum: |S| = ℏ√(s(s+1)), where ℏ is the reduced Planck constant. For electrons, s = ½, giving |S| = ℏ√(3/4).

The z-component of spin angular momentum is quantized as S_z = m_sℏ, where m_s can take values from -s to +s in integer steps. For electrons, m_s can be ±½.

The magnetic moment associated with spin is μ = -gμ_Bs, where g is the g-factor (≈2 for electrons) and μ_B is the Bohr magneton. The energy in a magnetic field is E = -μ·B.

Spin is fundamental to quantum mechanics and has important applications in magnetic resonance imaging (MRI), quantum computing, and understanding atomic and molecular spectra.

Key Concepts

  • Spin Magnitude: |S| = ℏ√(s(s+1))
  • Spin Projection: S_z = m_sℏ
  • Magnetic Moment: μ = -gμ_Bs
  • Energy in Field: E = -μ·B
  • Bohr Magneton: μ_B = eℏ/2m_e
  • G-Factor: g ≈ 2 (electron)

Real-World Applications

  • Magnetic Resonance Imaging (MRI)
  • Quantum Computing: Qubits
  • Nuclear Magnetic Resonance (NMR)
  • Electron Spin Resonance (ESR)
  • Atomic Spectroscopy

Physics Equations

Spin Magnitude:
∣S∣=ℏs(s+1)|S| = \hbar\sqrt{s(s+1)}
Spin Projection:
Sz=msℏS_z = m_s\hbar
Magnetic Moment:
μ=−gμBs\mu = -g\mu_B s
Energy in Field:
E=−μ⋅BE = -\mu \cdot B
Bohr Magneton:
μB=eℏ2me\mu_B = \frac{e\hbar}{2m_e}

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Spin Magnitude

The magnitude of spin angular momentum:

Equation:

∣S∣=ℏs(s+1)|S| = \hbar\sqrt{s(s+1)}

Calculation:

∣S∣=1.055e−34×0.5×(0.5+1)=9.133e−35 J\cdotps|S| = 1.055e-34 \times \sqrt{0.5 \times (0.5 + 1)} = 9.133e-35 \text{ J·s}

Explanation:

This gives the total magnitude of the spin angular momentum vector.

2

Step 2: Calculate Spin Projection

The z-component of spin angular momentum:

Equation:

Sz=msℏS_z = m_s\hbar

Calculation:

Sz=0.5×1.055e−34=5.273e−35 J\cdotpsS_z = 0.5 \times 1.055e-34 = 5.273e-35 \text{ J·s}

Explanation:

This gives the projection of spin along the z-axis.

3

Step 3: Calculate Magnetic Moment

The magnetic moment associated with spin:

Equation:

μ=−gμBs\mu = -g\mu_B s

Calculation:

μ=−2.002×9.274e−24×0.5=−9.285e−24 J/T\mu = -2.002 \times 9.274e-24 \times 0.5 = -9.285e-24 \text{ J/T}

Explanation:

This determines how the particle interacts with magnetic fields.

4

Step 4: Calculate Energy in Magnetic Field

The energy of the spin in the magnetic field:

Equation:

E=−μ⋅BE = -\mu \cdot B

Calculation:

E=−(−9.285e−24)×1.0=9.285e−24 JE = -(-9.285e-24) \times 1.0 = 9.285e-24 \text{ J}

Explanation:

This gives the energy difference between spin states in the field.

Frequently Asked Questions

What is quantum spin?

Quantum spin is an intrinsic form of angular momentum carried by particles. Unlike classical angular momentum, it's quantized and has no classical analogue.

What is the spin quantum number?

The spin quantum number s determines the magnitude of spin angular momentum. For electrons, s = ½, meaning the spin magnitude is ℏ√(3/4).

What are spin projections?

Spin projections m_s can take values from -s to +s in integer steps. For electrons (s = ½), m_s can be ±½, giving two possible spin states.

What is the magnetic moment?

The magnetic moment μ = -gμ_Bs is associated with spin and determines how the particle interacts with magnetic fields. For electrons, g ≈ 2.

How does spin relate to energy in a magnetic field?

The energy in a magnetic field is E = -μ·B. Different spin projections have different energies, leading to energy level splitting.

Practice MCQs

  1. For an electron (s = ½), the spin magnitude is:
  2. How many possible spin projections does an electron have?
  3. The magnetic moment of an electron is approximately:
  4. What happens to energy levels in a magnetic field?
  5. The Bohr magneton is given by: